function such that for all and almost every . Then, for all , The proof relies on the dominated convergence theorem and the mean value theorem (details below). Proofs Proof of basic form We first prove the case of constant limits of integration a and b. We use Fubini's theorem to change the order of integration. For every and , such that and both and are within , we have: Note that the integrals at hand are well defined since is continuous at the closed rectangle and thus also uniformly continuous there; thus its integrals by either dt or dx are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below). Therefore: Where we have defined: (we may replace x0 here by any other point between x0 and x) F is differentiable with derivative , so we can take the limit where approaches zero. For the left hand side this limit is: For the right hand side, we get: And we thus prove the desired result: Another proof using the bounded convergence theorem If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign. Note that this proof is weaker in the sense that it only shows that fx(x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx(x,t) (and thus is obviously also Lebesgue integrable). Let By the definition of the derivative, Substitute equation () into equation (). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so We now show that the limit can be passed through the integral sign. We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each δ > 0, consider the difference quotient For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that Continuity of fx(x, t) and compactness of the domain together imply that fx(x, t) is bounded. The above application of the mean value theorem therefore