I cannot figure out use $\varepsilon$-$\delta$ definition of continuity to do this question.

Question

Use the $\varepsilon$-$\delta$ definition of continuity to prove that the function $f: \mathbb{R} \longrightarrow \mathbb{R}$ given by $f(x) = x^2$ is continuous at $x = 1$.

First, I am having trouble understanding what is the $\varepsilon$-$\delta$ definition of continuity then how to use that to figure out how to start and finish the proof.

Answer 1

Hint: $|x^2-1|=|x+1||x-1|\leq (1+2|1|)\delta$ $\displaystyle \Rightarrow \delta<\frac{\varepsilon}{1+2|1|}=\frac{\varepsilon}{3}$