Sequentially Compact and composition

I had an exam question today along the lines of:

Prove or disprove: If such that the graph of f defined as is sequentially compact(something else was also sequentially compact, which I believe f was), then f must be continuous.

I do not know if this is true or false. Quite honestly, I can not picture this in my head. By counterexample of a characteristic function:

I would think to be false, but I don't think this can be rewritten as the composition of two functions.

Conversely, if f was continuous, then for h(x) = g(x,f(x)) would be continuous since it is the composition of continuous functions.

Thank you for reading. Any help is greatly appreciated.