In this problem we consider three functions $f$. The first two are continuous at $x=0$, i.e., $$\lim_{x\to 0}f(x) = f(0).$$ The third function is continuous from the right at $x=0$, $$\lim_{x \to 0^+} f(x) = f(0).$$
In order use the $\epsilon/\delta$ definition to prove the continuity statements, one must give a definition of $\delta$ in terms of $\epsilon$ such that $$0<|x-0| < \delta \quad \Longrightarrow \quad |f(x)-f(0)| < \epsilon.$$ To prove the right-continuity statement requires a definition of $\delta$ in terms of $\epsilon$ such that
$$0 < x-0 < \delta \quad \Longrightarrow \quad |f(x)-f(0)| < \epsilon.$$

For each function in the list below, enter the number (1,2, or 3) of one of these choices
1. $\delta = \epsilon^2$
2. $\delta = \epsilon$
3. $\delta = \sqrt{\epsilon}$
so that your choices establish continuity of the first two functions, and right-continuity of the third function, at $x = 0$. You may use each choice only once.

$f(x) = x$:
$f(x) = x^2$:
$f(x) = \sqrt{x}$: