Any non-constant function from the plane to {0,1} is of necessity discontinuous. If you compose such a function with a path on which the function is constant, then the result is a constant function from (an interval of) the reals to {0,1}, which is continuous. Here the y-axis lie...
A function f:X→Y is almost continuous in the sense of Stallings, if for each open set UX×Y containing f, U contains also a continuous function g:X→Y [J. Stallings, Fundam. Math. 47, 249-263 (1959; Zbl 0114.39102)]. See also T. Natkaniec [Real Anal. Exch. 17, No. 2, 46...
Given a function z = f(x, y), the function is continuous at points where the function is well-defined. So if a function has a square root, the function may not be defined for negative numbers and so on. Given these considerations, we look at a two-variable function and d...
are the simplest functions of a variablexthat are continuous for all values ofx. The sum, difference, and product of continuous functions again yield continuous functions. The quotient of two continuous functions is also a continuous function, except for those values ofxfor which the denominator va...
Absolutely continuous functions and random variables are related to each other in the following way: A real-valued random variable X is absolutely continuous if its distribution function FX is absolutely continuous [3]Formal DefinitionThe formal definition is frequently used in real analysis, ...
continuity, in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say x—is associated with a value of a dependent variable—say y. Continuity of a ...
check whether the function \(\begin{array}{l}\frac{4x^{2}-1}{2x-1}\end{array} \) is continuous or not? solution: at x=1/2 , the value of denominator is 0. so the function is discontinuous at x = 1/2 . definition of differentiability f ( x ) is said to be differentiable ...
limit definition a limit of a function is a number that a function reaches as the independent variable of the function reaches a given value. the value (say a) to which the function f(x) gets close arbitrarily as the value of the independent variable x becomes close arbitrarily to a ...
Set-valued functionTopological gamesBy means of topological games, we will show that under certain circumstances on topological spaces X, Y and Z, every two variable set-valued function F:X×Y→2Z is strongly upper (resp. lower) quasi-continuous provided that Fx is upper (resp. lower) semi...
Compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value f the independent variable. f(x) = 9x^2 - 2x + 8; x = 3 Why is Ampere's law not consistent with the equation of continuity? How was this set right ...