I am really unsure about my reading of this problem.

Is it correct that: “each of f and g is a continuous function and f agrees with g in a dense subset of the interval I in the real numbers”? If that is correct then the function f-g is zero on a dense subset. If f-g fails to be zero at some point then f-g fails to be continuous.

That is a rough outline of a proof based upon my reading of your problem.