$f(X)$ is differentiable at $x=a$ but it is discontinuous

Question

I have defined a function such that:

$$f(x)=\begin{cases}mx,&(x\le a), \\ mx+c,&(x>a).\end{cases}$$

Here according to the derivative definition :

$f '(a) = \lim_{x\to a} [f (x) - f (a) ]/ [x - a] $

we can show that this limit exits by taking the LHS and RHS limits , and showing that they are equal. Since the gradient is the same I think it is trivial .

Can anyone please explain ?

Thank you !

Answer 1

The right hand limit does not exist. Remember, $f(a) = ma$, not $ma + c$, even when you're finding the right hand limit.