What happens when numerically approximating the gradient of a function at a nondifferentiable point?

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I am asking myself this question: Can anything be said of the resulting gradient, when the gradient of a function is approximated numerically (for example using the finite differences method) at a nondifferentiable point?

Could this result in any number? For example, when considering a convex function, will this always approximate some subgradient?

I wasn't able to find anything on this question, is the answer totally obvious?