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).
Curves." §1.3.6 and 2.1.3 in