$\displaystyle f(x)=\sum_{n\geq1}{a_{n}\cos(nx)}+\sum_{n\geq1}{b_ {n}\sin(nx)}\ \ \ \ \ \ with \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}, \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}$
Use the Weierstrass M-test for the uniform convergence; and the theorem that a uniform limit of continuous functions is continuous for the continuity of f(x).