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Unformatted text preview: and g are continuous. Sketch the graphs of f and g . 7. Let f : R R be Dirichlets function dened by f ( x ) = 1 if x is rational , 0 if x is irrational . Prove that f is discontinuous at every point x R . 8. Let f : (0 , 1) R be dened as f ( x ) = if x is irrational , 1 q if x = p q in lowest terms . 2 (Recall that p q is in lowest terms if p and q are integers with no common factor and q &gt; 0.) Prove that f is continuous at every irrational x and discontinuous at every rational x . 9. Give an example of a function dened on R which is continuous at only one point and discontinuous at all the other points on R ....
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- Winter '10