= A derivative has no removable discontinuity
{title2=$\lim_{x\to c}f'(x)\text{ exists}\Longrightarrow\lim_{x\to c}f'(x)=f'(c)$}
Let $f$ be <differentiable> on an <open interval> containing $c$. If the finite <limit of a function> $L=\lim_{x\to c}f'(x)$ exists, then $f'(c)=L$.
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 that
$$
f'(x_n)=\frac{f(c+h_n)-f(c)}{h_n}.
$$
The right-hand side tends to $f'(c)$ 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 $f'$ 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>.
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