Is a Function Continuous in a Neighborhood If It's Continuous at a Point?

[JG89]

If f is continuous at x = a, then is it continuous in some neighborhood of x = a as well?

 

[lurflurf]

No
A standard example is
f(x)=x x rational
f(x)=0 x irrational

f is continuous at x=0

 

[Elucidus]

No
A standard example is
f(x)=x x rational
f(x)=0 x irrational

f is continuous at x=0

And nowhere else. Not even very close to 0.

--Elucidus