Continuity (Calculus)

If a function is continuous, then it has no ‘breaks’ in the curve. Continuity helps in finding out if a function is predictable and ‘correct.’

Definition §

Continuity

Let $f$ be a real-valued function (i.e. $f:D\to \mathbb{R}$). And let $a$ be a limit point of $D$. Then:

$$\text{f is continuous at a}\leftrightarrow \underset{x\to a}{\lim }f(x)=f(a)$$

Note, if $a$ is an isolated point, $f$ is always continuous at $a$, with no condition.

Theorem - Continuous Everywhere Functions §

The following function types are always continuous along their entire domain i.e. continuous everywhere:

• Polynomials
• trigonometric functions: sin x, cos x, tan x, sec x, cosec x, cot x, arcsin x, arccos x, arctan x
• exponential functions: $a^x,a>0$

• logarithm functions: ${\log }_a(x),x>0$
• $n$th root functions: $\sqrt[n]{x},n>1$
• absolute value function: |x|
• Hyperbolic functions

Theorem - Continuity Operations §

If the functions $f$ and $g$ are continuous at $x=a$, then the following functions are continuous (at $x=a$):

1. $f+g$,
2. $cf$ , where $c$ is a constant
3. $f\times g$,
4. $\frac{f}{g}$ if $g(a)\ne 0$

Theorem - Function Composition §

If $f$ is continuous at $x=a$ and $g$ is continuous at $x=f(a)$, then the Function Composition $f\circ g$ is continuous at $x=a$.

This theorem is very useful in finding continuity of complex functions by breaking them down into simpler functions and then checking which of them are continuous everywhere

Theorem - Continuity & Limits §

If a function can be represented as $f(g(x))$ and ${\lim }_{x\to a}g(x)=b$:

$$\underset{x\to a}{\lim }f(g(x))=f(\underset{x\to a}{\lim }g(x))=f(b)$$

• $f$ must be continuous at $b$.

This is very, very useful to break down Limit (Calculus), since we can simplify complex functions into those that are continuous everywhere, and then apply this theorem.

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• Differentiability
• Limits and Continuity of Bivariate Functions

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