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In my usage, "g(x)" is not acceleration -- ever, anywhere, anyhow. And, in
my opinion, such usage is extremely pedagogically detrimental. Depending on
the level of the class g is a local gravitational constant/factor. The
operative word here is "local" -- This usage doesn't require the value to
be the same everywhere on the face of the Earth for all time. (And if that
usage isn't free from dispute, then I will massage the definition until it
fits.) But others are free to use or misuse the language anyway they like.