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Homework Help: Sequences and continuity

  1. Nov 26, 2011 #1
    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?
     
  2. jcsd
  3. Nov 26, 2011 #2
    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?
     
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