The correct options are
A continuous' everywhere
C not differentiable at x = 0
E not differentiable at an infinite number of
points.
f(x)={1+sinx for sinx≥01−sinx for sinx<0
Clearly f(x) is continuous everywhere.
f′(x)={cosx for sinx≥0−cosx for sinx<0
f is differentiable everywhere except at the points where sinx=0,
i.e., x=nπ, nϵI. Hence f is also not differentiable at x=0.