# Continuity on an interval

1. Feb 2, 2009

### cesc

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

Suppose a function is continuous at a point, c. Does this mean there exists an interval around c which is also continuous?

If so prove

2. Relevant equations

3. The attempt at a solution

2. Feb 2, 2009

### Dick

Any opinion on whether it might be true or not? Doesn't seem true to me, but that's just an opinion also because I can't think why it would be. You might try to find a counterexample first.

3. Feb 5, 2009

### cesc

it boils down to the definition of the limit.

for all e>0. there exists s>0 such that

if x satisfies abs(x-a) then abs(f(x)-a)<e

the question is: Does f have to be defined on the interval abs(x-a)?

example of this- A function is undefined at every point except a.

does the limit exist at a?

if yes, then we have a trivial counterexample to the original post

4. Feb 5, 2009

### Dick

The counterexample isn't that trivial. Define f(x)=x if x is rational and f(x)=0 if x is irrational. Where is that continuous?

5. Feb 5, 2009

### cesc

ah, thanks for the counterexample.it would only be continuous at 0.

My second post was to clarify a technical point.
If f is undefined at every point except a, and defined at a, is f continuous at a?

6. Feb 5, 2009

### Dick

Continuity says as x->a, f(x)->f(a). If there are undefined points arbitrarily close to a, I would say no, it's not continuous. If you say the definition is x->a AND f(x) defined at x, then you could say yes, it is. A 'function' with 'undefined' points is a little ambiguous. In any event, even you decide to call it technically continuous, it's not a very interesting example, is it?