Differentiability implies continuity. I think you're supposed to show that f is continuous at x=0 and x=3. Not show that their derivatives exist at those points.

Yes since differentiability implies continuity, and the question asks to show that f is differentiable on R, it would be insufficient to show merely continuity (continuity does not imply differentiability). In my answer, I have shown that f is differentiable on R, thus it is certainly continuous on R. I do not have to repeat it again.