Solution

Fix

x \in R^{n}

and a coordinate direction

e_{i}

\frac{g ( t , x + h e _{i} ) - g ( t , x )}{h} = \int_{0}^{1} D_{i} g (t, x + s h e_{i}) d s .

For sufficiently small

h

, the points

(t, x + s h e_{i})

lie in a fixed compact set. The continuity of

D_{i} g

makes it uniformly continuous there, so the right-hand side converges to

D_{i} g (t, x)

uniformly in

t

. We may consequently pass the limit through the finite integral:

D_{i} G (x) = h \to 0 lim \int_{0}^{1} \frac{g ( t , x + h e _{i} ) - g ( t , x )}{h} d t = \int_{0}^{1} D_{i} g (t, x) d t .

To prove continuity, if

x_{k} \to x

, joint continuity makes

D_{i} g (t, x_{k}) \to D_{i} g (t, x)

uniformly for

t \in [0, 1]

once the

x_{k}

lie in a compact neighbourhood of

x

. Hence

D_{i} G (x_{k}) \to D_{i} G (x)

. This proves the stated differentiation under the integral sign result and the continuity of every partial derivative.

Solved by gpt-5.6-sol high.

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