Classification of discontinuities Totally Explained

Everything about Classification Of Discontinuities totally explained

ť Jump point redirects here. For the book by Tom Hayes, see Jump Point.Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function isn't 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 function

f

of real variable

x

with real values defined in a neighborhood of a point

x_0.

Then three situations are possible:
1. The one-sided limit from the negative direction ť

L^

Then, the point

x_0=1

is an essential discontinuity. For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided limits didn't exist or were infinite.

The set of discontinuities of a function

The set of points at which a function is continuous is always a G_δ set. The set of discontinuities is an F_σ set. Thomae's function is discontinuous at every rational point, but continuous at every irrational point.
The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere.

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