Question

Discuss the continuity of the function $f$ defined by $f\left(x\right)=\frac{1}{x},x\ne 0$.

Answer 1

Given $f\left(x\right)=\frac{1}{x}$

At $x=0$

$f\left(0\right)=\frac{1}{0}=\infty$

Hence $f\left(x\right)$ is not defined at $x=0$

By given $f\left(x\right)=\frac{1}{x},x\ne 0$

So we check for continuity except at $x=0$

'C' be any number except 0.

f is continuous at $x=C$

$\underset{x\to C}{lim}f\left(x\right)=f\left(C\right)$

$f\left(x\right)=\underset{x\to C}{lim}\frac{1}{x}=\frac{1}{C}$

$\therefore$ 'f' is continous at $x=C$ [Expect 0]

f is not continous for all $x\in R\sim \left\{0\right\}$