Thread: Continuous functions

Need some help on the following:

Suppose f and g are continuous real valued functions defined on I such that I is defined on the closed interval [0,1]. We also know that f(m/(2^n))=g(m/(2^n)) for all rational numbers in I. Prove that f=g.

Originally Posted by PauKelome

Suppose f and g are continuous real valued functions defined on I such that I is defined on the closed interval [0,1]. We also know that f(m/(2^n))=g(m/(2^n)) for all rational numbers in I. Prove that f=g.

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.