Discontinuities (first kind)

So, as we mentioned, in the graph of a function, jumps, gaps, sharp turns, corners, cusps are called discontinuities.

Think about it, it makes total sense. It's not continuous.

The first kind of discontinuity is called the point discontinuity, otherwise called a removable discontinuity.
As the name implies, it's a single point where the function "behaves weirdly" in a graph. In the terms of a limit, in a point or removable discontinuity, the limit for that function exists at that point but is not equal to the value of the function at that point.

Consider this graph of a function with a removable discontinuity:

Notice what happens as you get closer and closer to the value x = 1.
You get closer and closer to the value f(x) = y = 4.

BUT: When you have exactly x = 1, f(x) is 2.

So, the limit of the function as x approaches 1 exists and it's equal to 4. However, this limit is not equal to the value of the function at that point.

We say that this function has a point discontinuity/removable discontinuity/discontinuity of the first kind at x = 1.

Let's take a look at jump discontinuities, otherwise known as:
Discontinuities of The Second Kind