Chapter 2.3, Problem 61E

To determine

To show: The function $f\left(x\right)$ is continuous at $x=a$ if and only if $\underset{h\to 0}{\lim }f\left(a+h\right)=f\left(a\right)$.

Explanation of Solution

Given information: The function $f\left(x\right)$ is continuous at $x=a$.

Proof:

According to the definition of continuity, a function $f\left(x\right)$ is continuous at $x=a$ if and only if $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$.

Substitute $a+h$ for $x$ and if $x$ tends to $a$ then $h$ tends to $a$ . So, the expression is:

  $\begin{array}{c}\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)\\ \underset{h\to 0}{\lim }f\left(a+h\right)=f\left(a\right)\end{array}$

Hence, a function $f\left(x\right)$ is continuous at $x=a$ if and only if $\underset{h\to 0}{\lim }f\left(a+h\right)=f\left(a\right)$.