for G being RealNormSpace-Sequence
for F being RealNormSpace
for f being PartFunc of (product G),F
for X being Subset of (product G) st X is open holds
( ( for i being set st i in dom G holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )