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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 ′ exists. In other words, the graph of f has a non-vertical tangent line at the point (x 0, f(x 0)). f is said to be differentiable on U if it is differentiable at every point of U.
For z = 1/3, the inverse of the function x = 2 C 1/3 (y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, C z (y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero.
The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right. If f {\displaystyle f} is differentiable at a {\displaystyle a} , then f {\displaystyle f} must also be continuous at a {\displaystyle a} . [ 11 ]
This formula can fail when one of these conditions is not true. For example, consider g(x) = x 3. Its inverse is f(y) = y 1/3, which is not differentiable at zero. If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1/g′(f(0)). Since f(0) = 0 and g′(0) = 0, we must evaluate 1/0, which is ...
If there exists an m × n matrix A such that = + ‖ ‖ in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix , and the linear transformation that associates to the increment Δ x ∈ R n the vector A Δ x ∈ R m is, in this general setting ...
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.
The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0. In other words, the value of the constant function, y {\displaystyle y} , will not change as the value of x {\displaystyle x} increases or decreases.
A critical point of a function of a single real variable, f (x), is a value x 0 in the domain of f where f is not differentiable or its derivative is 0 (i.e. ′ =). [2] A critical value is the image under f of a critical point.