Continuity definition of a functional

Question

I'm having a hard time understanding the formal definition of continuity of a functional. I'm not sure if such questions are appreciated on this site; so let me know.

Definition: The functional $J[y]$ is said to be continuous at the point $\hat{y}\in linear function space$, if for any $\epsilon>0$, there is a $\delta>0$ such that $|J[y] - J[\hat{y}]|<\epsilon$, provided that $||y-\hat{y}|| < \delta$

I mean, for a given $\epsilon$, you can find the set of all functions $y$ for which $|J[y] - J[\hat{y}]|<\epsilon$. Can't you then always pick a large enough delta such that $||y-\hat{y}|| < \delta$, for all $y$ in this set?

Answer 1

It's very much the other way around. I give you the $\epsilon$, you find the $\delta$ for me. And remember, every $y$ that is within $\delta$ of $\hat y$ has to work.

Take for instance the function $J(y) = 2y$. I give you, say, $\epsilon=0.1$ and $\hat y=1$. Find a $\delta$ such that $J(y)$ is within 0.1 of $J(\hat y)=2$ for all $y$ within $\delta$ of $\hat y$. Obviously, $\delta=1$ (e.g.) will not work, since, for instance, $y=1.5$ is within the putative distance of $\hat y$, but the value of the function is not within the range I prescribed. For linear spaces other than R it works the same way.

Answer 2

Large $\delta$ means more $y$'s means that the $\cdots<\epsilon$ condition is harder to be true.

Answer 3

The definition says that $|J[y] - J[\hat{y}]| < \epsilon$ for all $y$ in the ball around $\hat{y}$ with radius $\delta$. Now, your statement has two problems:

First of all, such $\delta$ does not always exist, since your space can be unbounded.

Second, even if you have such a $\delta$, it is not guarantied that all points in this ball have the required property.