Explanation:
Classification of discontinuities
Continuous functions are of utmost importance
in mathematics and applications.
However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all
points of discontinuity of a function may be a discrete set, a dense set, or even the
entire domain of the function.
This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real
values.
Classification
of discontinuities
Consider a real valued function ƒ of
a real variable x, defined in a neighborhood of the point x_{0} in
which ƒ is discontinuous. Then three situations may be
distinguished:
1. The one-sided limit from the negative direction
and the one-sided limit from
the positive direction
at x_{0} exist, are finite, and are equal to L = L ^{−} = L ^{+} . Then, if ƒ(x_{0})
is not equal
to L, x_{0} is called a removable
discontinuity. This discontinuity can be 'removed
to make ƒ continuous at x_{0}',
or more precisely, the function
is continuous at x=x_{0}.
(Our solved example in mathguru.com uses this concept)
2.
The limits L ^{−} and L ^{+} exist and are finite, but not
equal. Then, x_{0} is called a jump discontinuity or step discontinuity. For this type of
discontinuity, the function ƒ may have any value in x_{0}.
3.
One or both of
the limits L ^{−} and L ^{+} does not exist or is infinite.
Then, x_{0} is called an essential discontinuity, or infinite discontinuity. (This is distinct from the
term essential singularity which is used when studying functions of complex
variables.)
http://en.wikipedia.org/wiki/Discontinuity_(mathematics)
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