f(x) is a continuous function. x is a J dimensional vector with each $x_j \in R$.
Is $f(x_{-j}) = f(x)$ thus f(x) is independent of $x_j$ if $\frac{d f(x_j)}{ d x_j}=0$.
Thus is a gradient that is always zero w.r.t to a certain input, enough for a continuous function to prove that its output value is independent of the specific input?
If this is true, how would I prove this mathematically?