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Continuity and differentiability This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity ). The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
Continuity of real functions is usually defined in terms of limits. A function f with variable x is continuous at the real number c, if the limit of (), as x tends to c, is equal to (). There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
The continuity equation says that if charge is moving out of a differential volume (i.e., divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore, the continuity equation amounts to a conservation of charge.
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]
An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical ...
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. A limit along a path may be defined by considering a parametrised path s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} in n-dimensional Euclidean space.