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The absolute value function is continuous (i.e. it has no gaps). 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 ...
The function f defined by f(0) = 0 and f(x) = x 3/2 sin(1/x) for 0<x≤1 gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.
If a continuous function on an open interval (,) satisfies the equality () =for all compactly supported smooth functions on (,), then is identically zero. [1] [2]Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", [2] since these weaker statements may be ...
The Weierstrass function has 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]
Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero. This is because that function, although continuous, is not differentiable at x = 0. The derivative of f changes its sign at x = 0, but without attaining the value 0.
If a function is semi-differentiable at a point a, it implies that it is continuous at a. The indicator function 1 [0,∞) is right differentiable at every real a, but discontinuous at zero (note that this indicator function is not left differentiable at zero).
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 : ′ = ′ = ′ (()).
A function of class is a function of smoothness at least k; that is, a function of class is a function that has a k th derivative that is continuous in its domain. A function of class or -function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that ...