Which of the following functions is differentiable at the origin?
E) None of the above

Question

Quizplus · Accepted Answer

The function represented by option B is of the form $f(x)=x\sin(\frac{1}{x})$ for $x 
eq 0$, and $f(0)=0$. It can be shown using the limit definition of derivative that $f'(0)=0$, hence the function is differentiable at the origin.

Option A is not differentiable at the origin since the limit defining the derivative does not exist. 
Option C is not even continuous at the origin, hence not differentiable at the origin. 
Option D is also not continuous at the origin, hence not differentiable at the origin.

Which of the Following Functions Is Differentiable at the Origin