(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x € A, then f(x) = 0 for all x € R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x € A, then f(x) = g(x) for all x € R. Hint. Ilse nart (a
(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x € A, then f(x) = 0 for all x € R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x € A, then f(x) = g(x) for all x € R. Hint. Ilse nart (a
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Transcribed Image Text:(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains
at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the
irrational numbers). Let A be a dense subset of R.
(a) Prove that if f is continuous and f(x) = 0 for all x E A, then f(x) = 0 for all x E R.
(b) Prove that if f and g are continuous functions and f(x) = g(x) for all x E A, then f(x) = g(x) for all x E R. Hint.
Use part (a).
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