You don't want to consider just "tangents" and, instead, consider what I defined as supporting hyperplanes of the epigraph and subgradients of the function. If the gradient exists, that is, if the function is differentiable, then the subgradient really is a tangent. Otherwise can have many different subgradients supporting at one point on the curve and its epigraph.

It's simple: A cube has supporting planes at each point that is an edge or corner, but those points do not have tangents.