
For functions that are "normal" enough, we know immediately whether or not they are continuous at a given point. Nevertheless, the continuity of a function is such an important property that we need a precise definition of continuity at a point:
Notes
Rather than returning to the $\varepsilon$$\delta$ definition whenever we want to prove a function is continuous at a point, we build up our collection of continuous functions by combining functions we know are continuous: If $f$ and $g$ are continuous at $c$, then
ExampleThe function $\displaystyle f(x)=\frac{x^24}{(x2)(x1)}$ is continuous everywhere except at $x=2$ and at $x=1$. The discontinuity at $x=2$ is removable, since $\displaystyle \frac{x^24}{(x2)(x1)}$ can be simplified to $\displaystyle \frac{x+2}{x1}$. To remove the discontinuity, define \[f(2)=\frac{2+2}{21}=4.\] We can also look at the composition $f\circ g$ of two functions, \[(f\circ g)(x)=f(g(x)).\] If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composition $f\circ g$ is continuous at $c$.
We'd also like to speak of continuity on a closed interval $[a,b]$. To deal with the endpoints $a$ and $b$, we define onesided continuity: A function $f$ is continuous from the left at $c$ if and only if $\displaystyle\lim_{x\to c^} f(x)=f(c)$. It is continuous from the right at $c$ if and only if $\displaystyle\lim_{x\to c^+} f(x)=f(c)$. We say that $f$ is continuous on $[a,b]$ if and only if
Note that $f$ is continuous at $c$ if and only if the right and lefthand limits exist and both equal $f(c)$.
Example
The function
\[
f(x)=\left\{\begin{array}{ll}
x, & x\leq 0\\
x^2, & 0 < x\leq 1\\
\frac{2}{x}, & 1 < x\leq 2\\
x1, & x > 2
\end{array}\right.
\]
is continuous everywhere except at $x=1$, where $f$ has a jump discontinuity.
Key ConceptsThat is, $f$ is continuous at $c$ if and only if for all $\varepsilon > 0$ there exists a $\delta > 0$ such that \[{\small\textrm{if }} xc < \delta {\small\textrm{ then }} f(x)f(c) < \varepsilon{\small\textrm{.}}\] In words, for $x$ close to $c$, $f(x)$ should be close to $f(c)$. 