Continuity

So what is continuity really?
For something to be continuous, it means to continue or behave as we would probably expect it to.

In terms of mathematics, this will mean the same, that the graph of a function will not have any sudden jumps, gaps, sharp turns, corners, cusps, etc...

These types of irregularities are called discontinuities, which are what you'll learn about in the next lesson. We still have some things to figure out.

Let's go back to that same graph of log(x) center

Notice that for every single point of this graph, any point you choose, for which log(x) exists and is defined, both the left and right-side limits exist. So, the general limit exists and is equal to the value of the function at that point.

This is exactly what it means for a function to be continuous:

A function is continuous at a point if the left and right limits for that point exist, so, the limit for that function exists at that point, and is equal to the value of the function at that point.

Along with the definition of a continuous function, this is probably even more important, remember this. Seriously.

All elementary functions are continuous on their domain.

Let's take a look at Discontinuities (first kind)