Continuous => limited in region

[Kruger]

Homework Statement

I've to show that if f:R->R is continuous in x' then f is limited in a suitable environment of x'.

2. The attempt at a solution

My lecturer said we should use the following inequality

|f(a)|=<|f(a)-f(y)|+|f(y)|

But how should I go on, I know I have to show something like this:

It exists d>0: it exist c element of [x'-d, x'+d]=I ==> |f(x)|=<|f(c)| for every x in I.

But how should I go on?

 

[StatusX]

Always include relevant definitions. "limited in a suitable environment" is not standard terminology, and I don't know what it means. It sounds like "bounded in some neighborhood", but this would follow imediately from the epsilon delta definition of continuity.

 

[Kruger]

I thought that, too. I mean that it follows imidiately, but our lecturer said that we've to take |f(a)|=<|f(a)-f(y)|+|f(y)| to show that.

With limited in a suitable environment it is ment what you said (bounded in some neightboorhood).

 

[StatusX]

Right, so pick any e>0, then a d>0 so that |y-a|<d => |f(y)-f(a)|<e, and then use that inequality to show there is an M with |f(y)|<M, all y in some neighborhood of a.