Question about continuous functions and the difference between the values of the function

Question

Is it true that if a function $f$ is continuous at a point $p$ then for ever $\epsilon > 0$ there exists $\delta > 0$ such that $$|x_1-p| < |x_2-p| < \delta \implies |f(x_1) - f(p)| < |f(x_2) - f(p)| < \epsilon?$$ That is, the closer the $x$s near $p$, then the closer the value of the function.

Answer 1

You are asking whether every function continuous at a point is monotone near that point. The answer is no, and $$ f(x) = \begin{cases} x\sin \frac1x, & \text{if } x\ne 0, \\ 0, & \text{if } x =0 \end{cases} $$ is a standard counterexample.