Find the values so the function is differentiable

Question

Struggling with this for some reason. I know that you have to check for continuity but I am confusing myself.

Find the values of a and b so that the following function is differentiable in $\mathbb{R}$. $$ f(x) = \begin{cases} -3x+a, & x\ne2 \\ b, & x=2 \end{cases}$$

Thanks in advance.

Answer 1

Use that $$f'(x_0)=\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$$ if this Limit exists and $h\neq 0$

Answer 2

For the sake of continuity we have $b=a-6$. In that case $f(x)=-3x+a$ for all real numbers $x$ and everywhere differentiable.