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For example, the function (pictured) = + is defined for all real numbers and is continuous at every such point. Thus, it is a continuous function. Thus, it is a continuous function. The question of continuity at x = − 2 {\displaystyle x=-2} does not arise since x = − 2 {\displaystyle x=-2} is not in the domain of y . {\displaystyle y.}
Continuity is a local property of a function — that is, a function is continuous, or not, at a particular point of the function domain , and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
For example, if in the mass continuity equation for flowing water, u is the water's velocity at each point, and ρ is the water's density at each point, then j would be the mass flux, also known as the material discharge. In a well-known example, the flux of electric charge is the electric current density.
A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, π/2), x 2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can
The image of the limit of a contains a single point f(x), so it does not contain the limit of b. In contrast, that function is lower hemicontinuous everywhere. For example, for any sequence a that converges to x, from the left or from the right, f(x) contains a single point, and there exists a corresponding sequence b that converges to f(x).
For example, in the classification of discontinuities: in a removable discontinuity , the distance that the value of the function is off by is the oscillation; in a jump discontinuity , the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
Every continuous function from a nonempty convex compact subset K of a Euclidean space to K itself has a fixed point. [9] An even more general form is better known under a different name: Schauder fixed point theorem Every continuous function from a nonempty convex compact subset K of a Banach space to K itself has a fixed point. [10]