False.
For example, the function f(x)=∣x∣ is continuous at x=0 :
x→0−limf(x)=0=x→0+limf(x)=>x→0limf(x)=0
f(0)=∣0∣=0=x→0limf(x) On the other hand,
h→0−limhf(0+h)−f(0)=h→0−limh∣h∣−0=−1
h→0+limhf(0+h)−f(0)=h→0+limh∣h∣−0=1
h→0−limhf(0+h)−f(0)=−1
=1=h→0+limhf(0+h)−f(0) Then h→0limhf(0+h)−f(0) does not exist.
Therefore the function f(x)=∣x∣ is not differentiable at x=0.
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