Chapter 3 - Continuous Functions, Smoothness and Limits, Continuity, Functions Without Jumps, Combining Continuous Functions, Fundamental Theorems of Continuity, Exponential and Trigonometric Functions

3.1 Real-Valued Functions: Mapping from Numbers to Numbers

Now we move from sequences of numbers to functions. Specifically, we’ll focus on real-valued functions, which take real numbers as input and produce real numbers as output. If $D$ is a set of real numbers, a real-valued function $f$ on $D$ is a mapping:

Think of a function as a machine: you put in a number $x$ from the domain $D$, and it spits out another number $f(x)$ in the real numbers.

The set of all real-valued functions on a domain $D$, denoted $R^D$, is itself a rich mathematical structure. We can do familiar algebraic operations with functions:

• Addition: If $f$ and $g$ are functions in $R^D$, their sum $(f+g)$ is a new function in $R^D$ defined by: $$(f+g)(x)=f(x)+g(x)$$
• Scalar Multiplication: If $f\in R^D$ and $\alpha \in \mathbb{R}$ is a scalar, then $(\alpha \cdot f)$ is a function in $R^D$ defined by: $$(\alpha \cdot f)(x)=\alpha \cdot f(x)$$
• Multiplication: The product of two functions $(f\cdot g)$ in $R^D$ is a function in $R^D$ defined by: $$(f\cdot g)(x)=f(x)\cdot g(x)$$

These operations make $R^D$ into a vector space over $\mathbb{R}$, and also something called an algebra.

Within $R^D$, we have special functions called constant functions. These functions always output the same value, no matter what input you give them. Two important constant functions are the zero function (0) and the one function (1):

• Zero Function (0): $0(x)=0$ for all $x\in D$
• One Function (1): $1(x)=1$ for all $x\in D$

These act as identities for addition and multiplication in $R^D$.

We can also define an order relation on $R^D$. We say $f\le g$ if for every $x$ in the domain $D$, $f(x)\le g(x)$. A function $f$ is non-negative if $0\le f$, meaning $f(x)\ge 0$ for all $x\in D$.

Monotone Functions: Functions that Don’t Change Direction

If the domain $D$ is a subset of the real numbers, we can talk about monotone functions, just like we talked about monotone sequences.

Definition 3.1.2: Monotone Functions on $\mathbb{R}$

Let $f:D\to \mathbb{R}$ where $D\subseteq \mathbb{R}$.

1. Monotonically Increasing (Non-decreasing): If for all $x,y\in D$, $x\le y⟹f(x)\le f(y)$.
2. Strictly Monotonically Increasing: If for all $x,y\in D$, $x<y⟹f(x)<f(y)$.
3. Monotonically Decreasing (Non-increasing): If for all $x,y\in D$, $x\le y⟹f(x)\ge f(y)$.
4. Strictly Monotonically Decreasing: If for all $x,y\in D$, $x<y⟹f(x)>f(y)$.
5. Monotone: If $f$ is either monotonically increasing or monotonically decreasing.
6. Strictly Monotone: If $f$ is either strictly monotonically increasing or strictly monotonically decreasing.

Monotone functions are well-behaved. They either always go “uphill” or always go “downhill” as you move from left to right in their domain.

Example: Consider $f(x)=x^n$ where $n$ is a natural number.

• If $n$ is odd (like $x,x^3,x^5,...$), $f(x)=x^n$ is strictly monotonically increasing on $\mathbb{R}$.
• If $n$ is even (like $x^2,x^4,x^6,...$), $f(x)=x^n$ is not monotone on $\mathbb{R}$. It’s decreasing for $x<0$ and increasing for $x>0$. However, it is monotonically increasing on $[0,\infty )$.

In the next sections, we’ll explore a crucial property of functions: continuity, which is essential for calculus and analysis.

3.2 Continuity: No Sudden Breaks

Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no sudden jumps, breaks, or holes in the graph. If you make a small change in the input $x$, the output $f(x)$ also changes only by a small amount.

Let’s make this intuitive idea precise with a formal definition. We’ll define continuity at a point first.

Definition 3.2.1: Continuity at a Point

Let $D\subseteq \mathbb{R}$ and $x_0\in D$. A function $f:D\to \mathbb{R}$ is continuous at $x_0$ if for every $ϵ>0$, there exists a $\delta >0$ such that for all $x\in D$, if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<ϵ$.

This definition mirrors the $ϵ$-$\delta$ definition of a limit. In fact, continuity at $x_0$ means that the limit of $f(x)$ as $x$ approaches $x_0$ exists and is equal to the function value at $x_0$.

Geometrically, this means that if you take a small interval around $x_0$ (of width $2\delta$), the function values $f(x)$ for $x$ in this interval will all lie within a small interval around $f(x_0)$ (of width $2ϵ$).

Definition 3.2.2: Continuity on a Domain

A function $f:D\to \mathbb{R}$ is continuous on $D$ (or simply continuous) if it is continuous at every point $x_0\in D$.

Examples of Continuous and Discontinuous Functions

Let’s look at some examples to understand continuity better.

Example 1: Polynomials and Absolute Value are Continuous

Example 3.2.3 (1): Polynomials are Continuous

For any natural number $n\ge 0$, the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^n$ is continuous on $\mathbb{R}$.

Example 3.2.3 (2): Absolute Value Function is Continuous

The function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=|x|$ is continuous on $\mathbb{R}$.

These functions have “smooth” graphs without any breaks.

Example 2: The Floor Function - Discontinuous Jumps

Example 3.2.3 (3): Floor Function is Discontinuous at Integers

The floor function $\lfloor x\rfloor :\mathbb{R}\to \mathbb{R}$ (which gives the greatest integer less than or equal to $x$) is discontinuous at every integer $x_0\in \mathbb{Z}$. It is continuous at every point $x_0\notin \mathbb{Z}$.

The floor function has jumps at integer values. As $x$ approaches an integer from the right, $\lfloor x\rfloor$ is constant, but as you approach from the left, it jumps down by 1.

Example 3: A Function Continuous Only at One Point - Wild Behavior

Example 3.2.3 (4): A Function Continuous Only at $x=1/2$

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined as: $f(x)=\{\begin{array}{ll}x& \text{if }x\in \mathbb{Q}\\ 1-x& \text{if }x\notin \mathbb{Q}\end{array}$ This function is continuous only at $x=1/2$. Everywhere else, it’s discontinuous. This shows that continuity can be a very delicate property.

Sequential Continuity: Linking Continuity and Sequences

There’s a useful way to characterize continuity using sequences. It connects the concept of continuous functions back to the sequences we studied in the previous chapter.

Theorem 3.2.4: Sequential Characterization of Continuity

Let $D\subseteq \mathbb{R}$, $x_0\in D$, and $f:D\to \mathbb{R}$. The function $f$ is continuous at $x_0$ if and only if for every sequence $(a_n{)}_{n\ge 1}$ in $D$ that converges to $x_0$ (i.e., ${\lim }_{n\to \infty }{a}_n=x_0$), the sequence $(f(a_n){)}_{n\ge 1}$ converges to $f(x_0)$ (i.e., ${\lim }_{n\to \infty }f(a_n)=f(x_0)$).

This theorem says: a function is continuous at a point if and only if it preserves limits of sequences approaching that point.

Proof of Sequential Characterization of Continuity (Sketch)

($\Rightarrow$) If $f$ is continuous at $x_0$, then it preserves sequential limits: Assume $f$ is continuous at $x_0$ and $(a_n)$ is a sequence in $D$ converging to $x_0$. Let $ϵ>0$. By continuity of $f$, there exists $\delta >0$ such that if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<ϵ$. Since $(a_n)\to x_0$, there exists $N$ such that for all $n>N$, $|a_n-x_0|<\delta$. Combining these, for $n>N$, $|f(a_n)-f(x_0)|<ϵ$. Thus $(f(a_n))\to f(x_0)$.

($\Leftarrow$) If $f$ preserves sequential limits, then $f$ is continuous at $x_0$ (Proof by Contradiction): Assume $f$ is not continuous at $x_0$. Then there exists some $ϵ>0$ such that for every $\delta >0$, there is an $x\in D$ with $|x-x_0|<\delta$ but $|f(x)-f(x_0)|\ge ϵ$. For each $n\in \mathbb{N}$, choose $\delta =\frac{1}{n}$, and find a point $a_n\in D$ with $|a_n-x_0|<\frac{1}{n}$ but $|f(a_n)-f(x_0)|\ge ϵ$. The sequence $(a_n)$ converges to $x_0$, but $(f(a_n))$ does not converge to $f(x_0)$ (because it’s always at least $ϵ$ away from $f(x_0)$), contradicting our assumption that $f$ preserves sequential limits.

This sequential characterization is often easier to use when proving properties of continuous functions. For example, it makes it easy to show that sums, products, and quotients of continuous functions are also continuous.

3.3 Algebraic Combinations of Continuous Functions: Building More Complex Continuity

Just like with limits of sequences, continuity is preserved under basic algebraic operations. This allows us to build up more complex continuous functions from simpler ones.

Corollary 3.2.5: Algebraic Continuity Laws

Let $D\subseteq \mathbb{R}$, $x_0\in D$, $\lambda \in \mathbb{R}$, and let $f:D\to \mathbb{R}$ and $g:D\to \mathbb{R}$ be functions continuous at $x_0$. Then:

1. Sum: The function $(f+g):D\to \mathbb{R}$ is continuous at $x_0$.
2. Scalar Multiple: The function $(\lambda \cdot f):D\to \mathbb{R}$ is continuous at $x_0$.
3. Product: The function $(f\cdot g):D\to \mathbb{R}$ is continuous at $x_0$.
4. Quotient: If $g(x_0)\ne 0$, then the function $(f/g):D^{\prime }\to \mathbb{R}$ is continuous at $x_0$, where $D^{\prime }=\{x\in D:g(x)\ne 0\}$.

We can prove these laws elegantly using the sequential characterization of continuity (Theorem 3.2.4) and the Limit Laws for sequences (Theorem 2.1.8).

Proof of Algebraic Continuity Laws (using Sequential Continuity)

We will use Theorem 3.2.4: $f$ is continuous at $x_0$ iff for every sequence $(a_n)\to x_0$, $(f(a_n))\to f(x_0)$.

1. Sum $(f+g)$: Let $(a_n)$ be a sequence in $D$ with ${\lim }_{n\to \infty }{a}_n=x_0$. Since $f$ and $g$ are continuous at $x_0$, we know ${\lim }_{n\to \infty }f(a_n)=f(x_0)$ and ${\lim }_{n\to \infty }g(a_n)=g(x_0)$. By the Sum Law for sequences (Theorem 2.1.8 (1)), ${\lim }_{n\to \infty }(f(a_n)+g(a_n))={\lim }_{n\to \infty }f(a_n)+{\lim }_{n\to \infty }g(a_n)=f(x_0)+g(x_0)$. Since $(f+g)(a_n)=f(a_n)+g(a_n)$ and $(f+g)(x_0)=f(x_0)+g(x_0)$, we have shown that for any sequence $(a_n)\to x_0$, $((f+g)(a_n))\to (f+g)(x_0)$. By Theorem 3.2.4, $(f+g)$ is continuous at $x_0$.

2. Scalar Multiple $(\lambda \cdot f)$: Similar proof using the Scalar Multiplication Law for sequences (Theorem 2.1.8 (2)).

3. Product $(f\cdot g)$: Similar proof using the Product Law for sequences (Theorem 2.1.8 (2)).

4. Quotient $(f/g)$: Assume $g(x_0)\ne 0$. Since $g$ is continuous at $x_0$, and $g(x_0)\ne 0$, we can show that there’s a neighborhood around $x_0$ where $g(x)\ne 0$. We restrict our domain to $D^{\prime }=\{x\in D:g(x)\ne 0\}$. Let $(a_n)$ be a sequence in $D^{\prime }$ with ${\lim }_{n\to \infty }{a}_n=x_0$. Since $g(x_0)\ne 0$ and $g$ is continuous, ${\lim }_{n\to \infty }g(a_n)=g(x_0)\ne 0$. By the Quotient Law for sequences (Theorem 2.1.8 (3)), ${\lim }_{n\to \infty }\frac{f(a_n)}{g(a_n)}=\frac{{\lim }_{n\to \infty }f(a_n)}{{\lim }_{n\to \infty }g(a_n)}=\frac{f(x_0)}{g(x_0)}$. Since $(f/g)(a_n)=\frac{f(a_n)}{g(a_n)}$ and $(f/g)(x_0)=\frac{f(x_0)}{g(x_0)}$, we have shown that for any sequence $(a_n)\to x_0$ in $D^{\prime }$, $((f/g)(a_n))\to (f/g)(x_0)$. By Theorem 3.2.4, $(f/g)$ is continuous at $x_0$.

Continuity of Polynomials and Rational Functions

Using these algebraic continuity laws, we can deduce the continuity of important classes of functions.

Corollary 3.2.7: Polynomials are Continuous

Polynomial functions $P:\mathbb{R}\to \mathbb{R}$ of the form $P(x)=a_n{x}^n+\cdots +a_0$ are continuous on $\mathbb{R}$.

Corollary 3.2.8: Rational Functions are Continuous (where defined)

Rational functions $R(x)=\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials and $Q\not\equiv 0$, are continuous on their domain, which is $\mathbb{R}\setminus \{x\in \mathbb{R}:Q(x)=0\}$ (all real numbers except the roots of $Q$).

Explanation:

• Polynomials: Start with the basic continuous function $f(x)=x$ (you can prove this directly from the $ϵ$-$\delta$ definition or sequential continuity). Constant functions are also continuous. Using the Product Law repeatedly, $x^2=x\cdot x$ is continuous, $x^3=x^2\cdot x$ is continuous, and so on. Using the Scalar Multiplication Law, $a_k{x}^k$ is continuous. Finally, using the Sum Law, the sum of continuous terms $a_n{x}^n+\cdots +a_0$ (a polynomial) is continuous.
• Rational Functions: Rational functions are quotients of polynomials. Since polynomials are continuous, and quotients of continuous functions are continuous where the denominator is non-zero (Quotient Law), rational functions are continuous everywhere except where the denominator polynomial is zero (i.e., at their poles).

These results give us a large family of continuous functions to work with, forming the building blocks for more advanced analysis. In the next sections, we’ll explore deeper properties of continuous functions, like the Intermediate Value Theorem and the Extreme Value Theorem.

3.4 The Intermediate Value Theorem (IVT): No Skipping Values

The Intermediate Value Theorem (IVT) is one of the most important consequences of continuity. It captures the intuitive idea that a continuous function cannot “skip” values. If a continuous function takes on two values, it must also take on all values in between.

Theorem 3.3.1: Intermediate Value Theorem (IVT)

Let $I\subseteq \mathbb{R}$ be an interval, and let $f:I\to \mathbb{R}$ be a continuous function. If $a,b\in I$ and $c$ is any number between $f(a)$ and $f(b)$ (i.e., $f(a)\le c\le f(b)$ or $f(b)\le c\le f(a)$), then there exists at least one number $z$ between $a$ and $b$ (i.e., $z\in [a,b]$ or $z\in [b,a]$) such that $f(z)=c$.

Imagine walking along a continuous path in the mountains. If you start at an elevation of $f(a)$ and end at an elevation of $f(b)$, you must pass through every elevation $c$ between $f(a)$ and $f(b)$ at some point along your path.

Proof of Intermediate Value Theorem (Using Supremum - for increasing case $f(a)\le c\le f(b)$)

Assume without loss of generality (O.E.d.A.) that $a\le b$ and $f(a)\le c\le f(b)$. Define the set $M=\{x\in [a,b]:f(x)\le c\}$.

Let $z=\sup M$ (supremum exists because $M$ is non-empty ($a\in M$) and bounded above (by $b$)). We want to show $f(z)=c$.

Proof by contradiction:

1. Assume $f(z)<c$: Since $f$ is continuous at $z$, for values slightly larger than $z$, $f(x)$ will still be close to $f(z)$ and thus less than $c$. This means we can find a point $z^{\prime }>z$ with $f(z^{\prime })<c$, so $z^{\prime }\in M$. But $z^{\prime }>z=\sup M$, contradiction.
2. Assume $f(z)>c$: Since $f$ is continuous at $z$, for values slightly smaller than $z$, $f(x)$ will still be close to $f(z)$ and thus greater than $c$. This means for some interval to the left of $z$, all $x$ in that interval have $f(x)>c$, so no points in that interval are in $M$. This contradicts the fact that $z=\sup M$ is the least upper bound (we could find a smaller upper bound).

Since both $f(z)<c$ and $f(z)>c$ lead to contradictions, we must have $f(z)=c$.

Application: Finding Roots of Polynomials

The IVT is very useful for proving the existence of roots (zeros) of equations.

Corollary 3.3.2: Existence of Roots for Odd-Degree Polynomials

Let $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0$ be a polynomial with $a_n\ne 0$ and odd degree $n$. Then $P$ has at least one real root (i.e., there exists $z\in \mathbb{R}$ such that $P(z)=0$).

Proof of Root Existence for Odd-Degree Polynomials

For large positive $x$, the term $a_n{x}^n$ dominates, and if $a_n>0$ and $n$ is odd, $P(x)\to +\infty$ as $x\to +\infty$. If $a_n<0$ and $n$ is odd, $P(x)\to -\infty$ as $x\to +\infty$. Similarly, as $x\to -\infty$, if $n$ is odd, $x^n$ changes sign, so $P(x)$ will go to $-\infty$ if $a_n>0$ and to $+\infty$ if $a_n<0$.

In either case, we can find a large positive value $N$ and a large negative value $-N$ such that $P(N)$ and $P(-N)$ have opposite signs (one positive, one negative). Since polynomials are continuous (Corollary 3.2.7), by the IVT, there must be some point $z$ in the interval $[-N,N]$ where $P(z)=0$.

3.5 The Min-Max Theorem (Extreme Value Theorem): Boundedness and Extremes

The Min-Max Theorem, also known as the Extreme Value Theorem (EVT), guarantees that a continuous function on a closed and bounded interval (a compact interval) attains both a maximum and a minimum value.

Theorem 3.4.5: Min-Max Theorem (Extreme Value Theorem)

Let $f:I=[a,b]\to \mathbb{R}$ be a continuous function on a compact interval $I=[a,b]$. Then there exist points $u,v\in [a,b]$ such that for all $x\in [a,b]$, $f(u)\le f(x)\le f(v)$. In other words, $f$ attains a minimum value $f(u)$ and a maximum value $f(v)$ on $[a,b]$.

This theorem is incredibly important for optimization problems. It guarantees that if you are looking for the maximum or minimum value of a continuous function on a closed interval, such values actually exist.

Proof of Min-Max Theorem (Existence of Maximum - Using Supremum)

We will prove the existence of a maximum value $v$. The existence of a minimum value $u$ is similar.

1. Boundedness: First, we need to show that $f$ is bounded on $[a,b]$. (Proof omitted here, but it involves assuming $f$ is unbounded and constructing a sequence that violates sequential continuity).

2. Existence of Supremum: Since $f$ is bounded above on $[a,b]$, the supremum of its values exists: $M=\sup \{f(x):x\in [a,b]\}$.

3. Finding a sequence approaching the supremum: By definition of supremum, for each $n\in \mathbb{N}$, there exists a point $x_n\in [a,b]$ such that $M-\frac{1}{n}<f(x_n)\le M$.

4. Using Bolzano-Weierstrass: The sequence $(x_n)$ is in the bounded interval $[a,b]$. By the Bolzano-Weierstrass Theorem, there exists a convergent subsequence $(x_{l(n)})$ converging to some point $v\in [a,b]$. Let $v={\lim }_{n\to \infty }{x}_{l(n)}$.

5. Using Continuity: Since $f$ is continuous at $v$ and $(x_{l(n)})\to v$, by sequential continuity, ${\lim }_{n\to \infty }f(x_{l(n)})=f(v)$.

6. Showing $f(v)=M$: From step 3, we know $M-\frac{1}{l(n)}<f(x_{l(n)})\le M$. Taking the limit as $n\to \infty$, we get $M\le f(v)\le M$. Thus, $f(v)=M$.

Therefore, the maximum value $M$ is actually attained by the function at some point $v\in [a,b]$.

Importance of Conditions:

It’s crucial to note that both continuity and compactness of the interval (closed and bounded) are necessary for the Min-Max Theorem to hold. If either condition is removed, the theorem may fail.

• Discontinuity: A discontinuous function on a closed interval may not attain a maximum or minimum (e.g., a function with a jump discontinuity).
• Non-compact Interval: A continuous function on a non-closed interval (like $(0,1]$) or an unbounded interval (like $[0,\infty )$) may also not attain a maximum or minimum (e.g., $f(x)=x$ on $[0,1)$ has no maximum).

These fundamental theorems – IVT and EVT – are cornerstones of analysis. They guarantee essential properties of continuous functions and provide powerful tools for solving problems in calculus, optimization, and many other areas of mathematics.

3.6 The Real Exponential Function: A Function That Grows Faster Than Anything

We’ve already encountered the exponential function $e^z={\sum }_{n=0}^{\infty }\frac{z^n}{n!}$ in the context of series and complex numbers. Now let’s focus on the real exponential function, $f(x)=e^x$, for real inputs $x\in \mathbb{R}$. We can define it using the power series:

This series converges absolutely for all real numbers $x$.

Theorem 3.6.1: Properties of the Real Exponential Function

The exponential function $\exp :\mathbb{R}\to (0,\infty )$, defined by $\exp (x)=e^x$, is strictly monotonically increasing, continuous, and bijective (one-to-one and onto) onto the positive real numbers $(0,\infty )$.

Let’s explore some key properties of the exponential function.

Always Positive and Greater Than 1 for Positive Inputs:

Corollary 3.6.2: Positivity of Exponential Function

For all $x\in \mathbb{R}$, $e^x>0$.

For all $x>0$, $e^x>1$.

This follows directly from the power series definition, as all terms are positive for $x\ge 0$, and the first term is 1. For negative $x$, we use the property $e^{-x}{e}^x=1$ (derived from Cauchy product of series).

Strictly Increasing:

Corollary 3.6.3: Monotonicity of Exponential Function

For all $z>y$, $e^z>e^y$.

If you increase the input, the output of the exponential function always increases. It’s always climbing uphill.

A Useful Inequality:

Corollary 3.6.4: Inequality for Exponential Function

For all $x\in \mathbb{R}$, $e^x\ge 1+x$.

This simple inequality is surprisingly useful in proofs and estimations.

Proof of $e^x\ge 1+x$

We use the limit definition $e^x={\lim }_{n\to \infty }(1+\frac{x}{n}{)}^n$. By Bernoulli’s Inequality (Lemma 2.2.7), for $n\in \mathbb{N}$ and $1+\frac{x}{n}>-1$ (which holds for sufficiently large $n$), we have $(1+\frac{x}{n}{)}^n\ge 1+n\cdot \frac{x}{n}=1+x$. Taking the limit as $n\to \infty$, we get $e^x={\lim }_{n\to \infty }(1+\frac{x}{n}{)}^n\ge 1+x$.

Continuity of the Exponential Function:

Proof of Continuity of Exponential Function (Theorem 3.6.1)

We prove continuity at $x=0$ first, then extend to all $x\in \mathbb{R}$.

Step 1: Continuity at $x=0$: For $-\frac{1}{2}\le x\le \frac{1}{2}$, using $e^x\ge 1+x$ and $e^{-x}\le \frac{1}{1+x}$, we get $1+x\le e^x\le \frac{1}{1-x}$ for $x\in [-\frac{1}{2},\frac{1}{2}]$. This implies $x\le e^x-e^0\le \frac{x}{1-x}$ and $|e^x-e^0|\le 2|x|$ for $x\in [-\frac{1}{2},\frac{1}{2}]$. Given $ϵ>0$, choose $\delta =\min (\frac{1}{2},\frac{ϵ}{2})$. Then for $|x|<\delta$, $|e^x-e^0|\le 2|x|<2\delta \le ϵ$. Thus $e^x$ is continuous at $x=0$.

Step 2: Continuity at arbitrary $x_0\in \mathbb{R}$: Write $e^x-e^{x_0}=e^{x_0}(e^{x-x_0}-e^0)$. For $ϵ>0$, choose $\delta =\min (\frac{1}{2},\frac{ϵ}{2e^{x_0}})$. Then for $|x-x_0|<\delta$, $|e^x-e^{x_0}|=e^{x_0}|e^{x-x_0}-e^0|<e^{x_0}\cdot 2|x-x_0|<e^{x_0}\cdot 2\delta \le ϵ$. Thus $e^x$ is continuous at any $x_0\in \mathbb{R}$.

3.7 The Natural Logarithm: The Inverse of Exponential

Since the exponential function $e^x$ is a bijective, strictly monotonically increasing, and continuous function from $\mathbb{R}$ to $(0,\infty )$, it has an inverse function, called the natural logarithm, denoted by $\ln (x)$ or $\log (x)$.

Corollary 3.6.5: Properties of the Natural Logarithm

The natural logarithm $\ln :(0,\infty )\to \mathbb{R}$ is a strictly monotonically increasing, continuous, bijective function. It satisfies the property:

$\ln (a\cdot b)=\ln a+\ln b\text{for all }a,b\in (0,\infty )$

The logarithm “undoes” exponentiation, and vice versa. If $y=e^x$, then $x=\ln y$.

Proof of Properties of Natural Logarithm

Follows from Theorem 3.6.1 and Theorem 3.5.3 (Inverse Function Theorem). The property $\ln (ab)=\ln a+\ln b$ is derived by using the property $e^{x+y}=e^x{e}^y$ and the fact that exp and ln are inverse functions.

3.8 Trigonometric Functions: Circles and Periodicity

We now introduce the trigonometric functions: sine ($\sin x$) and cosine ($\cos x$). We define them using power series, just like the exponential function:

These series converge absolutely for all complex numbers $z$, and therefore for all real numbers $x$.

Theorem 3.8.1: Continuity of Sine and Cosine

The functions $\sin :\mathbb{R}\to \mathbb{R}$ and $\cos :\mathbb{R}\to \mathbb{R}$ are continuous on $\mathbb{R}$.

Relationship to Complex Exponential:

A crucial link between trigonometric and exponential functions is given by Euler’s formula:

Theorem 3.8.2 (1): Euler's Formula

For all $z\in \mathbb{C}$: $e^{iz}=\cos (z)+i\sin (z)$

This formula beautifully connects complex exponentials to sines and cosines. It allows us to derive many trigonometric identities from properties of the exponential function.

Using Euler’s formula, we can express sine and cosine in terms of complex exponentials:

Theorem 3.8.2 (3): Sine and Cosine in Terms of Complex Exponentials

$\sin z=\frac{e^{iz}-e^{-iz}}{2i}$ $\cos z=\frac{e^{iz}+e^{-iz}}{2}$

These power series definitions and the connection to the complex exponential provide a rigorous foundation for trigonometric functions, allowing us to explore their properties analytically. We can derive trigonometric identities, analyze their periodicity, and understand their behavior using the tools of calculus and analysis.

This concludes our journey through Chapter 3. We’ve explored continuous functions, their algebraic properties, and fundamental theorems like the IVT and EVT. We’ve also introduced the exponential, logarithm, and trigonometric functions, laying the groundwork for differentiation and integration in the chapters to come.

Previous Chapter: Chapter 2 - Sequences and Series, Approaching Infinity, Divergence, Infinity, Algebra of Limits, Monotone Sequences, Weierstrass, Cauchy, Series, Summing Infinitely, Special Series and Operations

Next Chapter: Chapter 4 - Differentiable Functions, The Slope of a Curve, Rules for Differentiation, Mean Value Theorem and Beyond