A real function of a single real variable is differentiable at a point if its derivative exists there. For several real variables, a function , with an open set, is differentiable at iff there is a linear transformation such that

The linear transformation is the Fréchet derivative of at . Its matrix in the standard bases is the Jacobian (Lebl 2026, pp. 35-40).

Differentiability implies continuity and the existence of all directional derivatives, but the existence of all partial derivatives, or even all directional derivatives, does not imply differentiability. A sufficient condition is that all first partial derivatives exist in a neighborhood of and are continuous at (Lebl 2026, pp. 40-45, 48-49).

The limit in the definition allows to approach 0 along arbitrary paths. Writing , with and , the quotient must tend to 0 uniformly in . Checking separate limits along fixed directions is insufficient.

For complex functions, complex differentiability leads to the Cauchy-Riemann equations and the theory of holomorphic functions.

Amazingly, there exist continuous functions which are nowhere differentiable. Two examples are the blancmange function and Weierstrass function. Hermite (1893) is said to have opined, "I turn away with fright and horror from this lamentable evil of functions which do not have derivatives" (Kline 1990, p. 973).

See also

Analytic Function, Blancmange Function, Cauchy-Riemann Equations, Complex Differentiable, Continuous Function, Derivative, Directional Derivative, Fréchet Derivative, Holomorphic Function, Jacobian, Partial Derivative, Weierstrass Function

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References

Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford, England: Oxford University Press, 1990.Krantz, S. G. "Alternative Terminology for Holomorphic Functions" and "Differentiable and Curves." §1.3.6 and 2.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 16 and 21, 1999.Lebl, J. Basic Analysis II: Introduction to Real Analysis, Vol. II, version 6.3. pp. 35-45, 48-49, 2026. https://www.jirka.org/ra/realanal2.pdf.McMillan, T. C. "Examining Continuity, Partial Derivatives and Differentiability with Cylindrical Coordinates." College Math. J. 34, 11-14, 2003. https://doi.org/10.1080/07468342.2003.11921978.Trott, M. "Differentiation." §1.6.1 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Differentiable

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Weisstein, Eric W. "Differentiable." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Differentiable.html

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