The problem is divided into two questions:

${x}_{n}$ is a real valued sequence.

Prove that $f(\underset{n\to +\mathrm{\infty}}{lim\u2006sup}{x}_{n})$ = $\underset{n\to +\mathrm{\infty}}{lim\u2006sup}f({x}_{n})$ given that f is continuous and increasing.

What can we say when f is decreasing.