Q. 79 - Calculus

Question

In Exercises 69–80, determine whether or not f is continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

$f\left(x\right)=\left\{\begin{array}{ll}1,& \mathrm{if} x \mathrm{rational}\\ x+1,& \mathrm{if} x \mathrm{irrational}\end{array}\right.,x=1$

Step-by-Step Solution

Answer

The function $f\left(x\right)=\left\{\begin{array}{ll}1,& \mathrm{if} x \mathrm{rational}\\ x+1,& \mathrm{if} x \mathrm{irrational}\end{array}\right.,x=1$is not continuous and not differentiable at $x=1.$

Step 1. Given information.

The given function is $f\left(x\right)=\left\{\begin{array}{ll}1,& \mathrm{if} x \mathrm{rational}\\ x+1,& \mathrm{if} x \mathrm{irrational}\end{array}\right..$

The value of x is $x=1.$

Step 2. Graph of function.

Plot the graph of the function $f\left(x\right)=\left\{\begin{array}{ll}1,& \mathrm{if} x \mathrm{rational}\\ x+1,& \mathrm{if} x \mathrm{irrational}\end{array}\right..$

Step 3. Continuity of a function.

Between every two rational numbers, there is an irrational number so in the graph of the function $f\left(x\right)=1,$dotted line represents the infinity number of discontinuity.

Similarly, between every two irrational numbers, there is a rational number so in the graph of the function $f\left(x\right)=x+1,$dotted line represents the infinity number of discontinuity.

So the function is not continuous at $x=1$.

Step 4. Differentiability of function.

As every discontinuous function is not differentiable and the function $f\left(x\right)$is not continuous.

so the function is not differentiable.