Category: Continuity of the Function

Definition of Continuous Function

Definition. A function $f$ is continuous at ${a}$ if $\lim_{{{x}\to{a}}}={f{{\left({a}\right)}}}$.

Continuity implies three things:

1. ${f{{\left({a}\right)}}}$ is defined (i.e. ${a}$ is in the domain of ${f{}}$);
2. $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}$ exists;
3. $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}={f{{\left({a}\right)}}}$.

Geometrically, continuity means that you can draw a function without taking your pen off the paper.

One-Sided Continuity. Classification of Discontinuities

Similarly to the one-sided limits, we can define one-sided continuity.

Definition. Function ${f{{\left({x}\right)}}}$ is continuous from the right at point ${a}$ if $\lim_{{{x}\to{{a}}^{+}}}={f{{\left({a}\right)}}}$. Function ${f{{\left({x}\right)}}}$ is continuous from the left at point ${a}$ if $\lim_{{{x}\to{{a}}^{{-}}}}{f{{\left({x}\right)}}}={f{{\left({a}\right)}}}$.

Theorems involving Continuous Functions

Intermediate Value Theorem. Suppose that $f$ is continuous on closed interval ${\left[{a},{b}\right]}$ and let ${N}$ is any number between ${f{{\left({a}\right)}}}$ and ${f{{\left({b}\right)}}}$ (or ${f{{\left({b}\right)}}}$ and ${f{{\left({a}\right)}}}$; depends what is bigger). Then there exists number ${c}$ in ${\left({a},{b}\right)}$ such that ${f{{\left({c}\right)}}}={N}$.