Continuity and its definition

Question

Is it true that a function $f$ is continuous at point $a$ if the following holds? $$ \exists\varepsilon > 0 \ \ \ \exists\delta>0 \text{ such that }\ \ |x - a|<\delta \Rightarrow |\frac{f(x) - f(a)}{x - a}| < \varepsilon $$

I know that this is easily solved when we use the limit definition of derivatives, but can it be solved without using derivatives?

Answer 1

Take $$f(x)=\sqrt{|x|}$$

it is continuous at $x=0$ but

$$\frac{f(x)-f(0)}{x-0}$$ is not bounded near $0$.

Answer 2

Yes, it is true. In the younterval $(x-\delta,x+\delta)$, you have$$\bigl\lvert f(x)-f(a)\bigr\rvert\leqslant\varepsilon\lvert x-a\rvert.$$Therefore, $\lim_{x\to a} f(x)-f(a)=0$.