Given that

Checking continuity and differentiability at x = 0,

LHL:

LHL = f(x=0)

Hence, f is continuous at x = 0.

LHD at x =0,

RHD at x =0,

∵ LHD = RHD = f(0)

∴ f(x) is differentiable at x =0.

So, option A is correct.