Continuity

Definition of Continuity

A function $f(x)$ is considered continuous at a point $c$ if:

• $f(c)$ is defined.
• The limit $\lim_{x \to c} f(x)$ exists.
• The limit as $x$ approaches $c$ of $f(x)$ is equal to $f(c)$, i.e., $\lim_{x \to c} f(x) = f(c)$.

Continuity implies that $f(x)$ is continuous at every point in its domain. For an open interval $(a, b)$, $f(x)$ is continuous if it is continuous at every point $c$ where $a < c < b$.

Continuity at a Point

For $f(x)$ to be continuous at a point $x = a$, these conditions must be met:

• $f(a)$ exists.
• $\lim_{x \to a^-} f(x)$ exists and equals $f(a)$ (continuity from the left).
• $\lim_{x \to a^+} f(x)$ exists and equals $f(a)$ (continuity from the right).

Types of Continuity

• Continuity from the Right: $f(x)$ is continuous from the right at $a$ if $\lim_{x \to a^+} f(x) = f(a)$.
• Continuity from the Left: $f(x)$ is continuous from the left at $a$ if $\lim_{x \to a^-} f(x) = f(a)$.

Continuity on Closed Intervals

$f(x)$ is continuous on a closed interval $[a, b]$ if:

• It is continuous on $(a, b)$.
• It is continuous from the right at $a$ ($\lim_{x \to a^+} f(x) = f(a)$).
• It is continuous from the left at $b$ ($\lim_{x \to b^-} f(x) = f(b)$).

Discontinuities occur when these conditions are not met, which could mean $f(a)$ is undefined, the limit doesn't exist, or the limit as $x$ approaches $a$ doesn't equal $f(a)$.

Continuous Functions and Composition

If $f(x)$ is continuous at $b$ and $\lim_{x \to a} g(x) = b$ then the composition of functions $f(g(x))$ is continuous at $a$ and:

$$\lim_{x \to a} f(g(x)) = f\left( \lim_{x \to a} g(x) \right) = f(b)$$

This shows the relationship between the continuity of individual functions and their composition.

Examples of Continuous Functions

No.	Function	Continuity Condition
1	Polynomials	Continuous for all $x$
2	Rational functions	Continuous except where division by zero occurs
3	$\sqrt[n]{x}$ (n odd)	Continuous for all $x$
4	$\sqrt[n]{x}$ (n even)	Continuous for all $x \geq 0$
5	$e^x$	Continuous for all $x$
6	$\ln(x)$	Continuous for $x > 0$
7	$\cos(x)$ and $\sin(x)$	Continuous for all $x$
8	$\tan(x)$ and $\sec(x)$	Continuous except at $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \ldots$
9	$\cot(x)$ and $\csc(x)$	Continuous except at $x = 0, \pm\pi, \pm2\pi, \ldots$

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is an important concept in calculus and analysis which relates to the continuity of functions. It states the following:

• Given a function $f(x)$ that is continuous on the closed interval $[a, b]$, and let $M$ be any number between $f(a)$ and $f(b)$.
• There exists a number $c$ in the interval $(a, b)$ such that $f(c) = M$.

This theorem guarantees that if $f(x)$ is continuous over a range and takes on different values at the end of the interval, it will take on every value between them at least once. The IVT is fundamental for proving the existence of roots within an interval.

References

• calculus_cheat_sheet_limits | tutorial.math.lamar.edu