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It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable. If f is differentiable at a point x 0, then f must also be continuous at x 0. In particular, any differentiable function must ...
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone. The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ {\displaystyle \delta } depends on ε ...
A bump function is a smooth function with compact support.. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.
The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable (the size of a function domain interval over which function value differences are less than ) that depends on only , while in (ordinary) continuity there is a locally applicable that depends on both and . So uniform ...
Rather between any two points no matter how close, the function will not be monotone. The computation of the Hausdorff dimension of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that = + <.
If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points. If f is twice differentiable, then conversely, a critical point x of f can be analysed by considering the second derivative of f at x : if it is positive, x is a local minimum; if it is negative, x is a ...
For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f' has a (local) minimum or maximum.
If is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and is an analytic map, then is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.)