A derivative has no removable discontinuity (${\lim }_{x\to c}{f}^{\prime }(x)\text{ exists}⟹{\lim }_{x\to c}{f}^{\prime }(x)=f^{\prime }(c)$)

Let $f$ be differentiable on an open interval containing $c$. If the finite limit of a function $L={\lim }_{x\to c}{f}^{\prime }(x)$ exists, then $f^{\prime }(c)=L$.

Choose $h_n>0$ tending to zero with $[c,c+h_n]$ in the interval. The mean value theorem gives $x_n\in (c,c+h_n)$ such thatThe right-hand side tends to $f^{\prime }(c)$ by differentiability. Since $x_n\to c$ with $x_n\ne c$, the left-hand side tends to $L$. Uniqueness of the limit of a sequence gives equality. The same reasoning excludes an infinite limit of a function for $f^{\prime }$ at $c$.

Thus derivatives may be discontinuous, but a removable discontinuity cannot be repaired by changing the value of a derivative at a single point of differentiability.