Continuity

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 §

Let $f$ be a real-valued function (i.e. $f:D\to \mathbb{R}$). Then f is continuous at $x=a$ iff ${\lim }_{x\to a}f(x)=f(a)$ I.e. the limit as x approaches a is also equal to the value of f at x = a.

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: sinh x, cosh x, tanh x, sech x, cosech x, coth x, arcsinh x, arccosh x, arctanh x

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 $f\circ g$ 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 ${\lim }_{x\to a}g(x)=b$:

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

• $f$ must be continuous at $b$.

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