I came across the following question on math stack exchange: Differentiability at a point a implies continuity in a neighborhood of a , and I'm wondering how to prove the function, defined as $x^2$ when x is rational, and 0 when x is irrational, is differentiable at 0. I have had some exposure to epsilon delta proofs, but I have only used them on functions such as polynomial, exponential and trigonometric functions. I'm not show how to use one to show the function defined above, is differentiable at 0.
Thanks in advance.
Best Answer
Hint.
Consider the Newton quotient $$ g(x):=\frac{f(x)-f(0)}{x-0} $$
Note that $g(x)=x$ when $x\ne 0$ is rational and $g(x)=0$ when $x$ is irrational.
You want to show that $\lim_{x\to 0}g(x)=0$. Basically you want an inequality like $ |g(x)|<\epsilon $.