Continuity can be defined conceptually in a few different ways. A function
is continuous, for example, if its graph can be traced with a pen without
lifting the pen from the page. A function is continuous if its graph is an
unbroken curve; that is, the graph has no holes, gaps, or breaks. But terms
like "unbroken curve" and "gaps" aren't technical mathematical terms and at
best, only provide a reader with a description of continuity, not a definition.
The more formal definition of continuity is this: a function
f (x) is
continuous at a point
x = a, if and only if the following three conditions are
met. 1)
f (a) is defined. 2)
f (x) exists. 3)
f (x) = f (a). Otherwise, the function is discontinuous.
A function can be continuous at a point, continuous over a given interval, or
continuous everywhere. We have already defined continuity at a given point.
For a function to be continuous over an interval [a, b], that function must be
continuous at each point in the interval, as well as at both a and b. For a
function to be continuous everywhere, it must be continuous for every real
number.
Discontinuities in functions can be classified according to the reason that the
function is discontinuous at a given point. If there exists a vertical
asymptote at
x = a for a function, that function is said to have an infinite
discontinuity at
x = a.
Figure 3.1: The function f (x) = 
has an infinite discontinuity at
x = 1.
If
f (x)≠
f (x), then
f (x) does not exist, which means that the second condition necessary for
continuity is not met. Such a discontinuity is called a jump discontinuity,
and looks like this:
Figure 3.2: The function f (x) = - 1 for x < 0, f (x) = 1 for x≥ 0 has a
jump discontinuity at x = 0.
If
f (x)≠f (a), the function has what is often called a
point, removable, or gap discontinuity. The point
(a,f (x)) is often called a hole.
Figure 3.3: The function f (x) = 
has a point discontinuity
at x = 1.
