Let
R denote the set of all real numbers and let
f be a
function with
domain containing {
x ε
R |
a <
x <
b} and
range contained in
R and
a <
x0 <
b. Define a new function
Dx0(
f ) by
1. Domain Dx0( f ) = {x ε dom( f ) | x ≠ x0 }
2. for any x in its domain, Dx0( f ) = (f ( x ) - f ( x0 ) ) / ( x - x0 )
f is differentiable at x0 if there exists a function D*x0( f ) with domain containing ( a , b ) and range contained in R such that D*x0( f ) is continuous at x0 and D*x0( f ) = Dx0( f )
for all x ε dom( f ) such that x ≠ x0
Consequently, if f is differentiable, then f is continuous; however, the converse does not hold. That is, if f is continuous, f might not be differentiable. For example, the function
f ( x ) = Σ bn cos ( an πx )
for 1 < n < infinity, where a is odd, 0 < b < 1, and ab > 1 + 3π/2, due to Weierstraß, is everywhere continuous but nowhere differentiable.