(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if f: [a,b]→[itex]Re[/itex] is increasing and the range of f is a bounded interval then f is continuous.

2. Relevant equations

N/A

3. The attempt at a solution

I have no idea where to start, but I decided to start with a couple of things.

Proof: Let f: [a,b]→[itex]Re[/itex] increasing and the rg(f) be a bounded interval. Since f is increasing, there exists x,y[itex]\in[/itex][a,b] such that f(a)≤f(x)<f(y)≤f(b). Further, since [a,b] is a bounded interval, S={x|x[itex]\in[/itex][a,b]}, there's a sequence {x_{n}} such that lim[itex]_{n\rightarrow b}[/itex] x_{n}= f(b). Similarly for lim = f(a).

Now I'm not really sure where to go, or if I was even headed in the right direction.

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# Homework Help: Show f is continuous if the range of f is a bounded interval

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