Author(s): KC Border
The following is a reasonably useful condition for differentiating a Riemann integral. The proof may be found in Dieudonné [5, Theorem 8.11.2, p. 177]. One thing you have to realize is that for Dieudonné a partial derivative can be taken with respect to a vector variable. That is, if f : R n × R m where a typical element of R n × R m is denoted (x, z) with x ∈ R n and y ∈ R m. The partial derivative Dxf is a Fréchet derivative with respect to x holding z fixed.