Prove f(y) = y for every real number y

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The discussion focuses on proving that a continuous function f: R->R, which satisfies f(q) = q for all rational numbers q, also holds true for every real number y. The key point is to leverage the property that every irrational number can be approached by a sequence of rational numbers. By applying the continuity of the function, one can show that as the sequence of rational numbers converges to an irrational number, the function values must converge to that number as well. This leads to the conclusion that f(y) = y for all real numbers y. The proof hinges on the combination of continuity and the density of rational numbers in the real number line.
Annie B
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A function f: R->R is a continuous function such that f(q) = q for every rational number q.
Prove f(y) = y for every real number y.

I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?
 
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I moved your thread to our homework forums.

You can use exactly that property about limits, together with continuity.
 

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