Answer
Continuous if $b=1$.
Not differentiable at $x=0$.
Work Step by Step
Step 1. Evaluate the left and right hand limits of the function: $\lim_{x\to0^-}g(x)=0+b=b$ and $\lim_{x\to0^-+}g(x)=cos(0)=1$
Step 2. For the function to be continuous at $x=0$, we need to set $b=1$.
Step 3. For the function to be differentiable at $x=0$, the left and right derivatives need to be equal; we have $1=-sin(0)$, which does not stand. Thus, we can not make the function differentiable at $x=0$.
