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  2. Differentiable function - Wikipedia

    en.wikipedia.org/wiki/Differentiable_function

    In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

  3. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    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 ...

  4. Inverse function rule - Wikipedia

    en.wikipedia.org/wiki/Inverse_function_rule

    In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...

  5. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    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.

  6. Rolle's theorem - Wikipedia

    en.wikipedia.org/wiki/Rolle's_theorem

    the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the n th derivative exists on the open interval (a, b), and; there are n intervals given by a 1 < b 1 ≤ a 2 < b 2 ≤ ⋯ ≤ a n < b n in [a, b] such that f (a k) = f (b k) for every k from 1 to n. Then there is a number c in (a, b) such that the n ...

  7. Generalizations of the derivative - Wikipedia

    en.wikipedia.org/wiki/Generalizations_of_the...

    The q-derivative of a function is defined by the formula () = () (). For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation.

  8. Immersion (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Immersion_(mathematics)

    A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M × [0,1] → N such that for all t in [0, 1] the function H t : M → N defined by H t (x) = H(x, t) for all x ∈ M is an immersion, with H 0 = f, H 1 = g. A regular homotopy is thus a homotopy through immersions.

  9. Strict differentiability - Wikipedia

    en.wikipedia.org/wiki/Strict_differentiability

    In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both points used in the difference quotient to "move".