# Fundamental increment lemma

In single-variable differential calculus, the **fundamental increment lemma** is an immediate consequence of the definition of the derivative *f*Template:'(*a*) of a function *f* at a point *a*:

The lemma asserts that the existence of this derivative implies the existence of a function such that

for sufficiently small but non-zero *h*. For a proof, it suffices to define

and verify this meets the requirements.

## Differentiability in higher dimensions

In that the existence of uniquely characterises the number , the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose *f* maps some subset of to . Then *f* is said to be differentiable at **a** if there is a linear function

and a function

such that

for non-zero **h** sufficiently close to **0**. In this case, *M* is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of *f* at **a**. Notably, *M* is given by the Jacobian matrix of *f* evaluated at **a**.

## See also

## References

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