Differentiable

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).

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.