Continuous but Not Smooth

Since the function $ax+b$ and $\sin{2x}$ are both continuous and differentiable, $f(x)$ is continuous and differentiable if $f(x)$ is continuous and differentiable at $x=0$ .

For $f(x)$ continuous at $x=0$ ,

$\lim_{x \to 0^{+}} ax+b=\lim_{x \to 0^{-}} \sin{2x}$

$b=0$

Now we have $f(x)=ax$ for $x>0$ .

For $f(x)$ differentiable at $x=0$ ,

$\frac{d}{dx} ax+b=\frac{d}{dx} \sin{2x}$

$a=2\cos{2x}$

$a=2$

Hence, $a+b=2$ .