Continuity and Intervals: Exploring the Relationship

[cesc]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[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.

 

[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

 

[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?

 

[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?

 

[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?