Differentiability Implies Continuity
Theorem
If \(X \subseteq \mathbb{R}\) and \(f : X \to \mathbb{R}\) is differentiable at \(a\), then \(f\) is continuous at \(a\).
First consider:
\[\begin{align*}
\lim_{x \to x_0} (f(x_0) - f(x)) &= \lim_{x \to x_0} \left(\frac{f(x_0) - f(x)}{x_0 - x} (x_0 - x)\right) \\
&= \lim_{x \to x_0} \left(\frac{f(x_0) - f(x)}{x_0 - x}\right) \lim_{x \to x_0} (x_0 - x) \tag{1}\\
&= f'(x_0) \lim_{x \to x_0} (x_0 - x) \\
&= f'(x_0) \times 0 \\
&= 0 \\
\end{align*}\]
which implies that
\[\lim_{x \to x_0} f(x) = f(x_0).\]
The condition of differentiability allows the limit to be split at (1) since it exists.