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A differentiable function. In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point [1] [2] and exactly one inflection point. Properties
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .
Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by () = / is not differentiable at =. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.
A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its tangents: [7]: 69 () + ′ () for all and in the interval. A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity.
It states that if f is continuously differentiable, then around most points, the zero set of f looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of f. The circle, for instance, can be pasted together from the graphs of the two functions ± √ 1 - x 2.
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.
Thomae's function is a real-valued function of a ... and so is not differentiable at all irrational . ... is the graph of the restriction of to (,) ...