Question 39
Let $f : [0, 1] \to \mathbb{R}$ be a continuous function. We define the function $\tilde{f} : [0, 1] \to \mathbb{R}$ by
$$
\tilde{f}(x) = \begin{cases} \frac{1}{x} \cdot \int_{0}^{x} f(t) \, dt, & \text{if } x \gt 0, \\ f(0), & \text{if } x = 0. \end{cases}
$$
Show that:
a) the function $\tilde{f}$ is continuous in $0$ and differentiable on $(0, 1]$;
$$
\tilde{f}(x) = \begin{cases} \frac{1}{x} \cdot \int_{0}^{x} f(t) \, dt, & \text{if } x \gt 0, \\ f(0), & \text{if } x = 0. \end{cases}
$$
Show that:
a) the function $\tilde{f}$ is continuous in $0$ and differentiable on $(0, 1]$;