_Posted on September 26, 2024_ > [!definition] A function $f$ is continuous on an open interval containing $x$ if and only if $\large \lim_{t \rightarrow x} f(t) = f(x)$ > [!theorem] > **Theorem:** If $f$ is differentiable on $(a,b)$, then it is continuous. Proof: Suppose $f$ is differentiable on some open interval $(a,b)$. Then, for all $x \in (a,b)$, the limit $\large f'(x)=\lim_{t\rightarrow x} \frac{f(t)-f(x)}{t - x}$ converges. Note that we have $\large \lim_{t \rightarrow x} [f(t)-f(x)]=\lim_{t\rightarrow x} [(t-x)\frac{f(t)-f(x)}{t - x}] = \lim_{t\rightarrow x} [t-x] \lim_{t\rightarrow x}[\frac{f(t)-f(x)}{t - x}]=$ $\large 0f'=0.$ This implies that $\lim_{t\rightarrow x}f(t) = \lim_{t\rightarrow x} f(x) =f(x).$ This is the desired result. $\blacksquare$