Is h Continuous and Increasing?

[Rosey24]

Homework Statement

We have a worksheet with practice final questions and I'm really stuck on this one on continuity:

Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)\bigcap(0,1), h(x) <= h(y)

a) prove that if h is continuous on (0,1), then h is increasing.
b) Give a counterexample to show that this need not be true if h is not continuous.

Thanks so much for any help you can provide!

Homework Equations

The Attempt at a Solution

The Attempt at a Solution

 

[Dick]

Think about h(x)=1 for x in (0,1/2] and h(x)=0 in (1/2,1). Open boundaries make all the difference.

 

[HallsofIvy]

Dick's response is to part (b).

For (a), Suppose u< v in (0, 1). If h(u)> h(v), can you get a contradiction to "there exists a delta>0 such that for all y in (x, x+delta)(0,1), h(x) <= h(y)" using the intermediate value property?