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

$\displaystyle I = [a,b]$ and $\displaystyle f : I \to R$ such that

$\displaystyle \forall x \in I$, $\displaystyle f$ is bounded on a neighborhood $\displaystyle V_{\delta_{x}}(x)$ of x.

$\displaystyle J = (a,b)$ and $\displaystyle g : J \to R$ such that

$\displaystyle \forall x \in J$, $\displaystyle g$ is bounded on a neighborhood $\displaystyle V_{\delta_{x}}(x)$ of x.

Prove $\displaystyle f$ is bounded on $\displaystyle I$.

Give an example of a function $\displaystyle g$ that isn't bounded on $\displaystyle J$.

I have this question, and I don't know what $\displaystyle \delta_{x}$ means. I see that the endpoints satisfying the boundedness statement is the critical difference between f and g, making f bounded on $\displaystyle (a,b)$, but not requiring g to be. I have only worked with bounded continuous functions, and these functions aren't necessarily continuous.