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* <

*x*_{0} <

*b*. Define a new function

**D**_{x0}(

*f* ) by

1. Domain **D**_{x0}( *f* ) = {*x* ε **dom**( *f* ) | *x* ≠ *x*_{0} }

2. for any *x* in its domain, **D**_{x0}( *f* ) = (*f* ( *x* ) - *f* ( *x*_{0} ) ) / ( *x* - *x*_{0} )

*f* is **differentiable** at *x*_{0} 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 *x*_{0} and **D**^{*}_{x0}( *f* ) = **D**_{x0}( *f* )

for all *x* ε **dom**( *f* ) such that *x* ≠ *x*_{0}

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 *b*^{n} **cos** ( *a*^{n} π*x* )

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.