Answer to Question #303204 in Real Analysis for Sarita bartwal

Question

Give an example of each of the following:

i. a function with a removable discontinuity

ii. a totally discontinuous function

Answer 1

(i ) Example of a function with removable discontinuity

$f(x )= \begin{cases} \frac{x²-4}{x-2} &\text{if } x≠2\\ 1 &\text{if } x=0 \end{cases}$

Here f(x) is discontinuous at x=2. This discontinuity is removable discontinuity.

(ii) Example of a totally discontinuous function

$f(x) = \begin{cases} 1 &\text{if } x &\ is &\ rational &\ number \\ 0 &\text{if } x&\ is &\ ir rational &\ number \end{cases}$ This function is totally discontinuous function.