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Unformatted text preview: Math 410  Homework 3  Due Tuesday June 21
√
(1) Show that f (x) = x is diﬀerentiable at all points x0 in (0, ∞). What is the problem
at 0?
(2) For a natural number n ≥ 2 let
if x ≤ 0 n
x f (x) = 0 if x > 0. Show that f is continuous.
(3) For a natural number n let
if x = a c g ( x) = xn −an if x = a x−a Find the value of c which makes g continuous.
(4) Let f : R → R be any function for which f (x + y ) = f (x) + f (y ). Suppose also that
f is continuous and let c = f (1).
(a) For each natural number n prove that f (1/n) = c/n.
(b) Let x ∈ Q, i.e. x = p/q for integers p and q , and prove that f (x) = cx.
(c) Use the continuityof f to show that f (x) = cx for all x ∈ R.
(d) Is f also diﬀerentiable?
(5) A function is even if f (−x) = f (x) for all x and it is odd if f (−x) = −f (x) for all x.
Prove that if f is diﬀerentiable and odd then f is even.
(6) Prove that the equation x4 + 2x2 − 6x + 2 = 0 has exactly two solutions.
(7) Suppose that f : R → R has bounded derivative on R. Show that f is uniformly
continuous. Does the function f (x) = sin(x2 ) have bounded derivative? Is this
enough to exclude that f (x) = sin(x2 ) is not uniformly continuous? Is the function
√
g (x) = 1/ x for x > 1 uniformly continuous? ...
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 Summer '08
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 Math

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