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## Homework Statement

Let ##(a,b)\subset \mathbb{R}## be a bounded open interval. Prove that if ##f:(a,b)\to\ \mathbb{R}## is strictly increasing at each point in ##(a,b)##, then ##f## is strictly increasing on ##(a,b)##.

## Homework Equations

Let ##(a,b)\subset \mathbb{R}## be a bounded open interval in ##\mathbb{R}##. We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing at ##c\in(a,b)## if there is a ##\delta>0## such that both the following hold:

1) for each ##x\in(c-\delta,c)## we have ##f(x)<f(c)##

2) for each ##x\in(c,c+\delta)## we have ##f(c)<f(x)##.

We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing on ##(a,b)## if whenever ##x,y\in(a,b)## with ##x<y##, we find that ##f(x)<f(y)##.

## The Attempt at a Solution

I am not sure where to start here. This is a challenge problem in the section on the least upper bound, so I suppose it relates somehow, but I am not seeing the connection. Hints would be nice.

EDIT: I think I solved it.

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