Proof of Various Derivative Properties — Question 5

Question 4 - Solution

To show that $f$ is continuous at $x=a$, we must prove that $$\lim_{x\to a} f(x)=f(a).$$

Since $f$ is differentiable at $a$, by definition, $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.$$

Because this limit exists, the expression $$\frac{f(a+h)-f(a)}{h}$$ remains bounded and approaches a finite value as $h\to 0$.

Rewrite $f(a+h)-f(a)$ as $$f(a+h)-f(a) = \left(\frac{f(a+h)-f(a)}{h}\right)h.$$

Now take limits as $h\to 0$: $$\lim_{h\to 0}\bigl(f(a+h)-f(a)\bigr) = \left(\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\right) \left(\lim_{h\to 0} h\right).$$

Since $$\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}=f'(a) \quad\text{and}\quad \lim_{h\to 0}h=0,$$ we obtain $$\lim_{h\to 0}\bigl(f(a+h)-f(a)\bigr)=0.$$

Therefore, $$\lim_{h\to 0}f(a+h)=f(a).$$

This shows that $$\lim_{x\to a}f(x)=f(a),$$ so $f$ is continuous at $x=a$.