### The Definition of Continuity

We are looking for a mathematical definition which captures two ideas.

- The values of a function f(x) at points
**near a** are good predictors of the value of
f **at a**.
- The graph of f is a connected curve with no jumps, gaps, or holes.

The first statement is easily thought of in terms of limits. It says the values f(x) are close to f(**a**) for points x near **a**, and this is an imprecise way of saying that the limiting value of f(x) as x approaches **a** is f(**a**).

The second statement really says the same thing. Imagine drawing the graph of f. As your pen moves along the graph towards the point (**a**,f(**a**)) its height traces out the values of f(x) for x nearing **a**. To say that there is no hole or gap at x=**a** is to say that f(x) is defined at x=**a**. To say there is no jump at x=**a** is to say that the values of f(x) approach f(**a**) as x approaches **a**.

The following definition captures both statements 1 and 2 neatly and cleanly.

** Definition of Continuity **Let a be a point in the domain of the function f(x). Then f is **continuous at x=a** if and only if |

A function f(x) is **continuous on a set** if it is continuous at every point of the set. Finally, **f(x) is continuous** (without further modification) **if it is continuous at every point of its domain.** |

**Equipment Check 1:** The following is the graph of a continuous function g(t) whose domain is all real numbers.

Unfortunately, a goodly portion of the graph has been torn off. On the basis of what remains, what is g(7)?

**Equipment Check 2:** A function h(x) is defined for 0< x < 1. You can explore its values by typing in values for x and pressing the "Evaluate h(x)" button.