Examine the following functions for continuity. $f(x) = \frac {1} { x - 5}, x \: \neq \: 5 $

Question

$\begin{array}{1 1} \text{Yes, it is continuous} \\ \text{No, it is not continuous}\end{array} $

Answer 1

Toolbox:

• If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
• Sum, difference, product and quotient of any continuous functions is continous also.

Step 1:

$f(x)=\large\frac{1}{x-5}$

At $x=5$

$f(x)=\large\frac{1}{5-5}=\frac{1}{0}$

$\quad\quad\quad\quad\;\;\;=$Not defined.

Step 2:

When $x\neq 5\lim\limits_{\large x\to c}f(x)=\lim\limits_{\large x\to c}\large\frac{1}{x-5}=\frac{1}{c-5}$

$f(c)=\large\frac{1}{c-5}$

$f$ is continuous at $x\in R-\{5\}$