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Continuous functions

A- Show that if a function is continuous on all of R and equal to 0 at every rational point then it must be identically 0 on all of R
b- if f and g are continuous on all of R and f(r)=g(r) at every rational point,must f and g be the same function?

Solution Preview

a. Proof:
For any irrational point x, we can always find a sequence of rational points x_n such that x_n->x as n->oo. From the definition, ...

Solution Summary

There are two proofs regarding rational points and continuity of functions.