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.

Take a Study Break!