A real function is said to be differentiable at a point if its derivative exists at that point. The
notion of differentiability can also be extended to complex
functions (leading to the Cauchy-Riemann
equations and the theory of holomorphic functions ),
although a few additional subtleties arise in complex
differentiability that are not present in the real case.

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 ,

Holomorphic
Function ,

Partial Derivative ,

Weakly
Differentiable ,

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. Referenced
on Wolfram|Alpha Differentiable
Cite this as:
Weisstein, Eric W. "Differentiable." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Differentiable.html

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