Differentiable Implies Continuous

State the theorem, then write continuity $\lim_{x \to x_0} f(x) = f(x_0)$ in the equivalent form $\lim_{x \to x_0}(f(x) - f(x_0)) = 0$ that the proof checks.

Reproduce $\lim_{x \to x_0}(f(x) - f(x_0)) = \lim_{x \to x_0}\left[\frac{f(x) - f(x_0)}{x - x_0}\right](x - x_0) = f'(x_0) \cdot 0 = 0$, noting the first factor's limit is where differentiability is used.

Record that multiplying and dividing by $x - x_0$ is legal since the limit excludes $x = x_0$, so $x - x_0$ is small but never zero, and note this supports the product and quotient rules.