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) ̸= 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.