. That is, Then the momentum at any time t is found by integration: Now, the model situation of an instantaneous transfer of momentum requires taking the limit as , giving a result everywhere except at 0: Here the functions are thought of as useful approximations to the idea of instantaneous transfer of momentum. The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property which holds for all , should continue to hold in the limit. So, in the equation , it is understood that the limit is always taken outside the integral. In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if and are functions such that almost everywhere, then is integrable if and only if is integrable and the integrals of and are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions. History Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form: which is tantamount to the introduction of the δ-function in the form: Later, Augustin Cauchy expressed the theorem using exponentials: Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem). As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as where the δ-function is expressed as A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation