Finding if a piecewise function is differentiable

Question

Im trying to find if f(x) is differentiable for any value of m.

$$ f(x) = \begin{cases} 3x^2 + mx - 1, & x \le 1 \\ 2x+3, & x>1 \end{cases}$$

Do I check for continuity?

Thanks for the help in advance.

Answer 1

$f$ is continuous iff $m=3$. But it can’t obviously be differentiable for that $m$.

Answer 2

$f$ will be continuous iff $$ \lim_{x\to 1_{-}} (3x^2 + mx - 1)=\lim_{x\to 1_{+}} (2x+3)$$ so $$ 3+m-1 = 2+3\implies m= 3$$