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In Exercises 69-80, determine whether or not fis 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(x)=1x,x=0

Short Answer

Expert verified

The function is non-continuous and non-differentiable at x=0and graph is

Step by step solution

01

Step 1. Given information

Given functionf(x)=1x,x=0

02

Calculate the LHL and RHL and calculate

Calculating, we get

limx0-f(x)=limx0-1xlimx0-f(x)=10=undefined

Similarly

limx0+f(x)=limx0+1xlimx0+f(x)=10=undefined

f(0)=10=undefined

So the function is not continuous atx=0

03

Checking for differentiability

Checking, we get

limx0-f(x)-f(0)x-0=limx0-1x-10x-0=undefinedlimx0+f(x)-f(0)x-0=limx0+1x-10x-0=undefined

The function is not differentiable atx=0

04

Sketching the graph

The graph is

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