Thanks for the responses. Does this fact go for when we have gradients almost everywhere on the ball (and subgradients on the non-differentiable points)? Also, any citations or literature to go with this?

I wonder if there is also an example where all $x$ within the unit ball are defined for the function. I feel that this example only works (it is a valid counterexample) because there are undefined points in the ball.

Oh, thank you. I think you have answered my question quite well. I appreciate it.

I have read though it as well, and haven't been able to see how it answers my question either. Am currently reading "Representations and inequalities for generalized hypergeometric functions" by Dmitrii Karp. I think there may be some monotonic results that might help.

Aaron, I appreciate this so much. Thank you for your help.