advertisement

Analysis Homework #3 due Thursday, Nov. 25 1. Show that the function f defined by { 3x − 2 f (x) = 4x − 4 if x ≤ 2 if x > 2 } is continuous at all points. Hint: see Example 2.13 in your notes. 2. Show that the function f defined by { 2x + 1 f (x) = x+3 if x ≤ 1 if x > 1 } is not continuous at y = 1. Hint: show that the ε-δ definition fails when ε = 1. 3. Suppose f, g are continuous with f (x) = g(x) for all x ∈ Q. Show that f (x) = g(x) for all x ∈ R. Hint: assume f (y) ΜΈ= g(y) for some y and consider the case f (y) > g(y), the other case being similar; try to get a contradiction when ε = 12 [f (y) − g(y)]. 4. Show that the polynomial f (x) = xn + x − 1 has a root in (0, 1) for each n ∈ N. • You are going to work on these problems Friday in class. • When writing up solutions, write legibly and coherently. Use words, not just symbols. • Write your name and then MATHS/TP/TSM on the first page of your homework. • Your solutions may use any of the results stated in class (but nothing else). • NO LATE HOMEWORK WILL BE ACCEPTED.