Prove that if the function fis a nonconstant real-...

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  1. jonatã alves
    jonatã alves
    Since f is a non-constant function on there are two distinct real numbers x and ysuch that f(x)f(y). Without any loss of generality we mayassume that f(x)<f(y).
    Since is connected and f is continuous on and f(x), f(y) are two distinct points inthe range of f therefore, by Intermidiate value theorem f attainsevery value between f(x) and f(y). i.e., [f(x), f(y)] is containedin range of f.
    Since [f(x), f(y)] being an interval in it is uncountable and hence range of f isuncountable.

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