Why Does arctan Ensure Continuity in Sequences?

[Ted123]

If f is continuous function and (x_n) is a sequence then x_n \to x \implies f(x_n) \to f(x)
The converse f(x_n) \to f(x) \implies x_n \to x in general isn't true but why is it true, for example, if f is arctan?

 

[Ted123]

Let \mbox{arctan}(u_n) \to \arctan(u). Write x_n = \mbox{arctan}(u_n) and x = \mbox{arctan}(u), so x_n \to x. Now using x_n \to x \implies f(x_n) \to f(x), with f as tan gives the result. Why can you do this?