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
Consider a real valued function ƒ of
a real variable x, defined in a neighborhood of the point x0 in
which ƒ is discontinuous. Then three situations may be
1. The one-sided limit from the negative direction
and the one-sided limit from
the positive direction
at x0 exist, are finite, and are equal to L = L − = L + . Then, if ƒ(x0)
is not equal
to L, x0 is called a removable
discontinuity. This discontinuity can be 'removed
to make ƒ continuous at x0',
or more precisely, the function
is continuous at x=x0.
(Our solved example in mathguru.com uses this concept)
The limits L − and L + exist and are finite, but not
equal. Then, x0 is called a jump discontinuity or step discontinuity. For this type of
discontinuity, the function ƒ may have any value in x0.
One or both of
the limits L − and L + does not exist or is infinite.
Then, x0 is called an essential discontinuity, or infinite discontinuity. (This is distinct from the
term essential singularity which is used when studying functions of complex
The above explanation is copied from
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