Cesaro Mean Limit Without Pointwise Limit

a) Exhibit a continuous function $f: (0, \infty) \to \mathbb{R}$ such that $$ \lim_{x \to \infty} \frac{1}{x^2} \int_0^x f(t) \, dt = 1, $$ but $f(x)/x$ has not a limit as $x \to \infty$. b) Let $f: (0, \infty) \to \mathbb{R}$ be an increasing function such that $$ \lim_{x \to \infty} \frac{1}{x^2} \int_0^x f(t) \, dt = 1. $$