Q. 80 - 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.

$f\left(x\right)=\left\{\begin{array}{ll}{x}^2,& \mathrm{if} x \mathrm{rational}\\ 2x-1,& \mathrm{if} x \mathrm{irrational}\end{array}\right.,x=1$

Step-by-Step Solution

Answer

 The function $f\left(x\right)=\left\{\begin{array}{ll}{x}^2,& \mathrm{if} x \mathrm{rational}\\ 2x-1,& \mathrm{if} x \mathrm{irrational}\end{array}\right.$is continuous and differentiable at $x=1.$

Step 1. Given information.

 The given function is $f\left(x\right)=\left\{\begin{array}{ll}{x}^2,& \mathrm{if} x \mathrm{rational}\\ 2x-1,& \mathrm{if} x \mathrm{irrational}\end{array}\right..$

The given value of x is $x=1.$

Step 2. Graph of function.

Plot the graph of the function.

Step 3. Continuity of a function.

Graph of function $f\left(x\right)=x^2$state that the value of the function is approaching to $1.$

Graph of function $f\left(x\right)=2x-1$state that the value of the function is approaching to $1.$

The value of the function $f\left(x\right)=1$at $x=1.$

The point of intersection of both functions is $(1,1).$ 

so the function $f\left(x\right)$is continuous at $x=1.$

Step 4. Differentiability of function.

Differentiate the function

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}{x}^2,& \mathrm{if} x \mathrm{rational}\\ 2x-1,& \mathrm{if} x \mathrm{irrational}\end{array}\right. \\ f\text{'}\left(x\right)=\left\{\begin{array}{ll}2x,& \mathrm{if} x \mathrm{rational}\\ 2,& \mathrm{if} x \mathrm{irrational}\end{array}\right. \end{gathered}$$

The first derivative f' will be continuous at the point of intersection.

$$\begin{gathered} 2x=2 \\ x=1 \end{gathered}$$

first derivative f' is continuous at $x=1.$

So the function f is differentiable at $x=1.$