The limit of a function as $x$ approaches $a$ can often be found simply by calculating the value of the function at $a$. Functions having this property are called continuous at $a$. We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language.
A function $f$ is continuous at a number $a$ if
$$ \lim_{x\rarr a}f(x)=f(a) $$
Notice that this requires three things if $f$ is continuous at $a$:
$$ f(a) \text{ is defined} $$
$$ \lim_{x\rarr a}f(x) \text{ exists} $$
$$ \lim_{x\rarr a}f(x)=f(a) $$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as $x$ approaches $a$.
Thus a continuous function $f$ has the property that a small change in $x$ produces only a small change in $f(x)$. In fact, the change in $f(x)$ can be kept as small as we please by keeping the change in $x$ sufficiently small.
If $f$ is defined on an open interval containing $a$, except perhaps at $a$, we say that $f$ is discontinuous at $a$ if $f$ is not continuous at $a$.
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The graph of a function $f$. At which numbers is f discontinuous? Why?
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Where are each of the following functions discontinuous?
$$ f(x)=\frac{x^2-x-2}{x-2} $$
$$ f(x) = \begin{cases} \frac{x^2-x-2}{x-2} & \text{if } x \ne2\\ 1 & \text{if } x = 2\\ \end{cases} $$
$$ f(x) = \begin{cases} \frac{1}{x^2} & \text{if } x \ne0\\ 1 & \text{if } x = 0\\ \end{cases} $$
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A function $f$ is continuous on an interval if it is continuous at every number in the interval. (If $f$ is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
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Show that the function $f(x)$ is continuous on the interval $[-1, 1]$.
$$ f(x)=1-\sqrt{1-x^2} $$
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Instead of always using Definitions to verify the continuity of a function, it is often convenient to use the next theorem, which shows how to build up complicated continuous functions from simple ones.
If $f$ and $g$ are continuous at $a$ and $c$ is a constant, then the following functions are also continuous at $a$:
$$ \text{1. }f+g $$
$$ \text{4. }fg $$
$$ \text{2. }f-g $$
$$ \text{5. }\frac{f}{g}\text{ if } \ g(a)\ne0 $$
$$ \text{3. }cf $$