Exploring the Relationship between Sequences and Continuity: The Case of arctan

[Ted123]

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

 

[Ted123]

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