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The value of the function at a critical point is a critical value. [1] More specifically, when dealing with functions of a real variable, a critical point, also known as a stationary point, is a point in the domain of the function where the function derivative is equal to zero (or where the function is not differentiable). [2]
A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x 0 is an interior point in the domain of a function f , then f is said to be differentiable at x 0 if the derivative f ′ ( x 0 ) {\displaystyle f'(x_{0})} exists.
A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function () = | |, at a = 0.
In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. [13] Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points ...
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (− r ) = f ( r ) , Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero.
The Weierstrass function has been historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. [1]
The global extrema of a function f on a domain A occur only at boundaries, non-differentiable points, and stationary points. If is a global extremum of f, then one of the following is true: boundary: is in the boundary of A; non-differentiable: f is not differentiable at
A convex function (blue) and "subtangent lines" at (red). In mathematics, subderivatives (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. [1]