(All Steps) A) Let f(x) = 1 for rational numbers x and f(x) = 0 for irrational numbers. Show that f is discontinuous at every x in . Hint: Let x . Select
Question:
A) Let f(x) = 1 for rational numbers x and f(x) = 0 for irrational numbers. Show that f is discontinuous at every x in
.
Hint:
Let x
. Select a sequence (x
n
) such that lim x
n
= x,
x n is rational for even n, and x n is irrational for odd n. Then f (x n ) is 1 for even n and 0 for odd n,
so (f (x n )) cannot converge.
B) Let h(x) = x for rational numbers x and h(x) = 0 for irrational numbers. Show that h is continuous at x = 0 and at no other point.
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[Steps Shown] Suppose that f is a real-valued continuous function
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[All Steps] Let f(x) = x 2 for x rational and f(x) = 0 for x irrational.
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