If derivative of $f$ is continuous, then $f$ is continuous.
If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.
In particular, even if $f'$ is discontinuous, $f$ is continuous.
Your problem seems to be the logical relationships between the statements
- If f is differentiable, then it is continuous
- If the derivative of $f$ is continuous, then $f$ is continuous
- If the derivative of $f$ is not continuous, then $f$ is not continous.
The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{\prime}$ is continuous".
The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function $$f(x)=x^2\sin\left(\frac{1}{x}\right)$$ is differentiable everywhere, with derivative $$f^{\prime}(x)=\left\{\begin{array}{ll} 2x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)& x\neq 0 \\ 0 & x=0 \end{array}\right.$$ But $\lim_{x\to 0}f^{\prime}(x)$ does not exist, hence $f^{\prime}$ is not continuous.
$f'$ need not be continuous.
Suppose that $f'(x)$ exists in the interval $(a,b)$. If $\xi \in (a,b)$, then $f'(\xi)$ exists. Hence $f$ is continuous at $\xi$. Since this is true for all $\xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.