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is everywhere continuous. However, it is not differentiable at = (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable. The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable.
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 ...
The Weierstrass function is continuous everywhere but differentiable nowhere. In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are ...
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.
It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be countably infinite or finite. According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a ...
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative; one direction follows from the mean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is ...
A mixed random variable does not have a cumulative distribution function that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue. The likelihood of a customer experiencing a zero wait time is discrete, while non-zero wait times are evaluated on a continuous time scale. [16]