Question

Discuss the continuity and differentiability at x=0

Answer 1

Answer:

f(x)=∣x∣+∣x−1∣

It can be written as

f(x)=

⎩

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎧

−2x+1;

+1;

2x−1

x<0

0≤x<1

x≥1

At x=0

LHL=−2(0)+1=1

RHL=1

LHD=−2

RHD=0

Continuous but not differentiable on x=0

At x=1 LHL=2(1)−1=1

RHL=1

LHD=1

RHD=2

Continuous but not differentiable on x=1

Continuous everywhere in (−1,2) but not differentiable on x=0, 1 in (−1,2).