# Difficult continuity question

1. Aug 15, 2007

### Rosey24

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

3. The attempt at a solution

2. Aug 15, 2007

### 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.

3. Aug 16, 2007

### HallsofIvy

Staff Emeritus
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?