Let $\mathcal{H} \subset C^0([0,1]; \mathbb{R})$ be a compact set.
Define, for some positive integer $n$, a mapping $g_n : \mathcal{H} \to \mathbb{R}$ as $$g_n(f) = \Big | \int_{0}^1 f(x) \cos(2 \pi n x) \, dx \Big | $$
Show that the above mapping is continuous.
For every $f \in \mathcal{H}$, for every $\epsilon > 0$, there exists a $\delta(\epsilon, f)$ so that for all $h \in \mathcal{H}$ such that $||f - h||_\infty < \delta(\epsilon, f)$, we have that
$$\Bigg | \Big | \int_{0}^1 f(x) \cos(2 \pi n x) \, dx \Big | - \Big | \int_{0}^1 h(x) \cos(2 \pi n x) \, dx \Big | \Bigg | < \epsilon$$
Note: $||f - h||_\infty = \sup \limits_{x \in [0,1]} |f(x) - h(x)|$
EDIT: Choose $\delta(\epsilon, f) = \frac{\pi}{2} \epsilon$. It works.