Question

• Show that if a function is differentiable for an interval, it is continuous over that interval.
• A function is continuous if \(\displaystyle\lim_{x\to a}f(x)=f(a)\).

Solution

Note that the derivative of \(f(x)\) is defined by $$f’(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$

Now we are trying to go from \(\displaystyle\lim_{x\to a}f(x)\) to \(f(a)\).

$$ \begin{align*} \lim_{x\to a}f(x)&=\lim_{x\to a}(f(x)-f(a)+f(a))\\ &=\lim_{x\to a}\left(\frac{f(x)-f(a)}{x-a}(x-a)+f(a)\right)\\ &=\lim_{x\to a}\left(\frac{f(x)-f(a)}{x-a}\right)\lim_{x\to a}(x-a)+\lim_{x\to a}f(a)\\ &=f’(a)\cdot0+f(a)\\ &=f(a) \end{align*} $$

We can reach the last step only if \(f’(a)\) is defined. In other words, if \(f\) is differentiable at \(a\), we can guarantee that it’s continuous at \(a\).

If \(f\) is differentiable for all points in the interval, then it is also continuous for all points in that interval.