Search results
Results from the WOW.Com Content Network
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
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.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set .
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
Closed graph theorem [5] — If : is a map from a topological space into a Hausdorff space, then the graph of is closed if : is continuous. The converse is true when Y {\displaystyle Y} is compact .
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 Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. [10] Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space.
The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ( π x ) 2 n {\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}} takes the value 1 {\displaystyle 1} when x {\displaystyle x} is an integer and 0 {\displaystyle ...