If f is bounded on every neighborhood of x in [a,b], then f is bounded on [a,b]

and such that

, is bounded on a neighborhood of x.

and such that

, is bounded on a neighborhood of x.

Prove is bounded on .

Give an example of a function that isn't bounded on .

I have this question, and I don't know what means. I see that the endpoints satisfying the boundedness statement is the critical difference between f and g, making f bounded on , but not requiring g to be. I have only worked with bounded continuous functions, and these functions aren't necessarily continuous.