Continuity of a function

 A function $f(x)$ is said to be continuous at $x=a$ if $f(x)$ converges to $f(a)$ as $x\to a$. Continuous functions are, in a sense, well-behaved and hence, easy to handle. 

Definition (Continuous function)

Let $f(x)$ be a function defined on an interval $I$ such that $a\in I$. The function $f(x)$ is said to be continuous at $a$ if $$\underset{x\to a}{lim}f(x)=f(a).$$ $f(x)$ is said to be a continuous function if it is continuous at every $x\in I$.

Remark. According to the definition of limits, $f(x)$ is continuous at $x=a$ if the following condition is satisfied:

• For any $\epsilon >0$, there exists $\delta >0$ such that, for all $x\in \mathrm{dom}f$, $0<|x-a|<\delta$ implies $|f(x)-f(a)|<\epsilon$.

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Example. For the function $f(x)$ defined by

$$f(x)=\{\begin{array}{cc}0& (x<0),\\ 1& (x\ge 0),\end{array}$$

$\underset{x\to 0}{lim}f(x)$ does not exist. Therefore it is not continuous at $x=0$. □

Example. For the function $f(x)$ defined by

$$f(x)=\{\begin{array}{cc}|x|& (x\ne 0),\\ -1& (x=0),\end{array}$$

we have $\underset{x\to 0}{lim}f(x)=0$. However, $f(0)=-1\ne \underset{x\to 0}{lim}f(x)$. Therefore it is not continuous at $x=0$. □

Remark. If the function $f(x)$ is defined on a closed interval $[a,b]$, we say it is continuous at $x=a$ if

$$\underset{x\to a+0}{lim}f(x)=f(a),$$

[Note the right limit], and continuous at $x=b$ if

$$\underset{x\to b-0}{lim}f(x)=f(b)$$

[Note the left limit]. □

Theorem (Properties of continuous functions)

Let $f(x)$ and $g(x)$ be functions that are continuous at $x=a$. Then the following functions are also continuous at $x=a$.

1. $kf(x)+lg(x)$ where $k,l$ are constants,
2. $f(x)g(x)$,
3. $\frac{f(x)}{g(x)}$ provided that $g(a)\ne 0$.

Proof. Trivial from the definition of continuous functions and the properties of limits. ■

Example. Any polynomial functions are continuous at all $a\in \mathbb{R}$. □

Theorem (Continuity of a composite function)

Let $f(x)$ be a function defined on an interval including $x=a$ such that $b=f(a)$. Let $g(x)$ be a function defined on an interval including $x=b$. If $f(x)$ is continuous at $x=a$ and $g(x)$ is continuous at $x=b$, then $(g\circ f)(x)$ is continuous at $x=a$.  

Proof. Trivial from the definition of continuity and limit of a composite function. ■

We give the following theorems without proof, but they show some of the important properties of continuous functions. Note that both theorems require the function to be defined on a closed interval.

Theorem (Intermediate Value Theorem)

Let $f(x)$ be a continuous function defined on a closed interval $[a,b]$ such that $f(a)\ne f(b)$. Then, for any $l$ between $f(a)$ and $f(b)$, there exists a $c\in [a,b]$ such that $f(c)=l$.

See also: Intermediate Value Theorem (Wikipedia)

Theorem (Extreme Value Theorem)

A continuous function $f(x)$ defined on a closed interval $[a,b]$ has a maximum value and a minimum value, each at least once.

See also: Extreme Value Theorem (Wikipedia)

Example. Consider the function

$$f(x)=\frac{1}{1-x^2}$$

defined on $(-1,1)$. This is a continuous function. However since $\underset{x\to 1-0}{lim}f(x)=+\mathrm{\infty }$ and $\underset{x\to -1+0}{lim}f(x)=+\mathrm{\infty }$ so it has neither a maximum nor a minimum. □