In your explorations of functions and their graphs, you will have noticed that the graphs of some functions, from start to end, are made of one unbroken curve, whereas others include breaks within their domain.

The following graphs demonstrate this idea.

Functions that contain no breaks along their entire domain are continuous.

As the \(x\)-values of a continuous function vary, the \(y\)-values vary continuously and do not jump from one value to another.

If a function is continuous, then it is possible to draw the function's graph from one end of its domain to the other without removing your pencil from the paper.

If this is not possible, then the graph must contain breaks.

A function that is not continuous at \(x=a\) is referred to as discontinuous at \(a\).

The point, \(a\), is known as a point of discontinuity.

A function, \(f(x)\), is continuous at \(x=a\) if \(\displaystyle \lim_{x\to a} f(x)=f(a)\).

For this definition to be satisfied, the following three conditions must hold.