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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. The oscillation of a function at a point quantifies these discontinuities as follows: in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point. This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ...
Point plot on the interval (0,1 ... is the graph of the restriction of to (,) ... function which is continuous on the rational numbers and discontinuous on the ...
Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some > such that for every >, we can find a point such that | | < and | () |.
The Dirichlet function is not Riemann-integrable on any segment of despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
Discontinuous linear map; Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions; Open mapping theorem (functional analysis) – Condition for a linear operator to be open; Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)