# hw3 - Math 410 - Homework 3 - Due Tuesday June 21 √ (1)...

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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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