Continuity and Limits
Continuity
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 .
has an infinite discontinuity at
x = 1
.
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:
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.
has a point discontinuity
at
x = 1
.
