What is continuity and how do we determine if a function is continuous or discontinuous at a point on its graph?

A function is said to be continuous at a point if all of the following are true:

The function's -value approaches some number as approaches from the left;

The function's -value approaches this same number as approaches from the right;

, the function's value at, is equal to .

If one or more of these requirements is not met, we say the function is discontinuous at , or that there is a discontinuity at .
What do we mean by approaches from the right or left? Suppose that we want to examine the graph at for continuity. Drag the red dot on the -axis so that it is placed at . We want to observe what happens along the graph as "sneaks up" on from the left. Check only the "Show L" box on the right side of the app. A small blue dot to the left of the red appears on the -axis. This is our "sneaking" value. The left slider at the right of the app will allow you to slide the blue dot closer and closer to the red dot (). Follow the blue dotted line up to the graph to see the point for that -value, then follow the arrow to the -axis to see the -value. When the blue dot is as close to the red dot as you can make it, we say we have reached the limit as approaches from the left, of , whose value is the -value there, . We would write this as . Notice the "" following ; it indicates "from the negative side (the left)".
Now check the "Show R" box and repeat, approaching from the right this time. Again, it appears that approaches . In this case, went to the same value regardless of which way we approached . But once we get to exactly , there is no point waiting for us; only an empty hole. We would write this approach as . Notice the "" following ; it indicates "from the positive side (the right)".
We will look at three types of discontinuity:
A removable discontinuity exists when the function approaches the same -value () from both sides of , BUT there is NO point connecting the two there. There is a "hole" in the graph - as if someone plucked out a single point from an otherwise continuous graph. It is called a removable discontinuity because "filling the hole" fixes the discontinuity. This is done by adding the definition that . In the graph above, a removable discontinuity exists at , which could be removed (filled in) by letting .
A jump discontinuity exists when the function approaches different and finite-values from the two sides of . In the graph above, there is a jump discontinuity at , where approaches from the left, but approaches from the right, as approaches .
An infinite discontinuity exists when the function approaches or on either or both sides of . In the graph above, there is an infinite discontinuity at , because approaches from both the right and from the left. As long as is approaching on at least one side, it is an infinite discontinuity.
If all three requirements are met, we can say that the function approaches the same -value from both sides of , AND there is a point connecting the two. The function is continuous at . This is true for all other points on the graph not mentioned above.