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 %: The function
*f* (*x*) =
has an infinite discontinuity at
*x* = 1
.

Figure %: The function
*f* (*x*) = - 1
for
*x* < 0
,
*f* (*x*) = 1
for
*x*≥ 0
has a
jump discontinuity at
*x* = 0
.

Figure %: The function
*f* (*x*) =
has a point discontinuity
at
*x* = 1
.