Simple analysis continuity problem

[economist1985]

Homework Statement

If f is a real function which is continuous at a element R and if f(a)<M for some M element of R, prove that there is an open interval I containing a such that f(x)<M for all x element of I.

Homework Equations

Extreme value theorem, intermediate value theorem, definition of continuity

The Attempt at a Solution

I have no idea how to solve this.

 

[snipez90]

Draw out the scenario. You have a continuous function with some point a on the x-axis, f(a) on the y-axis. Draw a horizontal line through the point f(a). We can think of M as a horizontal line somewhere above f(a). Let D be the distance between f(a) and M. If we can make f(x) within D of f(a), then M will be a bound for f(x) as well. But this is guaranteed by the continuity of f (epsilon-delta definition).

 

[economist1985]

Thanks for the explanation. The problem is clearer now to me. What I still do not get is how the epsilon delta definition guarantees it. ?? Sorry if I can't follow through... I'm new to this.

 

[economist1985]

Oh I think I got it. Thanks