Why would $f(x)$ at point $b$ not equal $f(b)$?

Question

In the picture below, I am told that points a and b represent different kinds of discontinuities.

Point a is discontinuous since the point limit of $f(x)$ as $x$ tends to $a$ doesn't exist. That does sound quite cryptic, my interpretation is that since $f(x)$ doesn't tend to $a$ when $x<a$, that point can't be regarded as continuous.

I'm more confused about point $b$ though. The book states that since this point cannot be continiuous since $A\neq f(b)$.

I am having a hard time comprehending this statement, surely at point $b$, the function $f(x)=f(b)$ by definition?

Answer 1

Yes, obviously $f(x)=f(b)$. However, the limit of the function when $x\to b$ is not $f(b)$. As you can see the values of $f$ near the point $x=b$ (not at the point itself but at its neighborhood) are very far from $f(b)$.

Answer 2

The point $f(b)$ is the black dot.

However, it is not equal to $\lim_{x \to b} f(x)=A \ne f(b)$, hence it is not continous at $b$.

The points around $b$ takes values that are closer to $A$ and far from $f(b)$.