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