Continuity and the Intermediate Value Theorem

The following theorems give us an easy way to determine if a complicated function is continuous. We simply break the function into simpler functions, and if our function is the sum/difference/product/quotient/composition of continuous functions, then it is continuous.

Theorem: If

$f(x)$ and

$g(x)$ are continuous at

$x=a$ , and if

$c$ is a constant, then

$f(x)+g(x)$ ,

$f(x)-g(x)$ ,

$cf(x)$ ,

$f(x)g(x)$ , and

$\frac{f(x)}{g(x)}$ (if

$g(a)\ne 0$ ) are continuous at

$x=a$ .

In short: the sum, difference, constant multiple, product and quotient of continuous functions are continuous.
(to understand why, see * below)

Theorem: If

$f(x)$ is continuous at

$x=b$ , and if

$\displaystyle{\lim_{x \to a} g(x) = b}$ , then

$\displaystyle{\lim_{x \to a} f(g(x)) = f(b)}$ .

In short: the composition of continuous functions is continuous.
(to understand why, see ** below)

Theorem: polynomial, rational, root, trigonometric, inverse trigonometric, exponential, and logarithmic functions are continuous at every number in their domain.

Why do we care? By definition, if $f$ is continuous at $x=a$ , then $\displaystyle{\lim_{x\rightarrow a}f(x)=f(a)}$ , which (if you think about it for a minute, and please DO) means that to evaluate the limit of a continuous function $f$ as $x\to a$ , you need only plug in $a$ to $f$ . It is easy to take the limit of a continuous function!

The following video goes over these properties and how to use them.

* This theorem is a direct result of limit laws. For instance, to see that

$f(x)+g(x)$ is continuous at

$x=a$ , we need to show that

$\displaystyle{\lim_{x \to a} (f(x)+g(x))}=f(a)+g(a)$ .
But

$\begin{array}{lll} \displaystyle\lim_{x\to a} f(x) &= f(a) &\hbox{(since $f(x)$ is continuous),}\\ \displaystyle\lim_{x\to a} g(x) & = g(a) & \hbox{(since $g(x)$ is continuous),} \\ \displaystyle\lim_{x \to a} (f(x) + g(x)) &\displaystyle= \lim_{x \to a} f(x) + \lim_{x \to a} g(x) & \hbox{(by our limit laws), so}\\ \displaystyle\lim_{x \to a} (f(x) + g(x)) &= f(a) + g(a), \end{array}$
as required. Deriving the other properties is similar.

** To see this, suppose that $x$ is close to (but not equal) to $a$ . Then $g(x)$ is close to $b$ , since $\displaystyle{\lim_{x \to a}g(x)=b}$ . Let $y=g(x)$ . Since $f$ is continuous at $b$ , whenever $y$ is close to $b$ , $f(y)$ is close to $f(b)$ . But that makes $f(g(x))$ close to $f(b)$ .

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

ff vs. f′f'

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Continuity properties

$f$ vs. $f'$