Find all values of the constants a and b for which the following function is differentiable:

Question

anonymous · Answer

$$f(x)= \begin{cases} 
      x^2+4x+1, & x\geq 1 \
      ax+b, & x < 1 \
      \end{cases}$$

I understand (I hope not to have misunderstood) that for a function to be a differentiable one at a certain point, its derivative coming from the left must be the same to that coming from the right. So in this case:

$$\large{\lim_{x\to 0^-}f’(x) = \lim_{x\rightarrow 0^+} f’(x)}$$
$$a= 2x + 4$$
$$a= 2(1) + 4$$
$$a= 6$$

By making the derivative at $x=1$ equal when coming from both sides, the function becomes a continuous one as well.

Since the derivative of a constant is zero, does it mean that $b$ can be any number?

Can a function be differentiable but not continuous?

Sorry for the silly questions.