The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Show that the function $H$ defined on $\mathbb{R}$ by: $$H(x) = \int_{0}^{x} h(t) \, dt$$ is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.
The function $h$ is defined on $\mathbb{R}$ by
$$h(u) = u - [u] - 1/2$$
Show that the function $H$ defined on $\mathbb{R}$ by:
$$H(x) = \int_{0}^{x} h(t) \, dt$$
is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.