Let be a continuous function on . Let for be equicontinuous on [0,1]. I.e., such that if , then for all n.
What can we conclude about ?
All I am able to get is that since each is defined on a compact set, then each is pointwise bounded. So, is uniformly bounded. So, is uniformly bounded.
Is there something else that I can conclude?
Actually, I am having a bit of difficult getting showing it is constant. I consider and let . Now I suppose such that . Now let be a natural number larger than y. It follows that .
However, I am having troubles with the 's in this case. for a given epsilon, Does the intermediate value theorem guarantee that I can find such that and ?