Math 214
Homework for Section 0.5
Spring 2016
Problem 1. Let R+ = {x ∈ R : x ≥ 0}.
1. Prove that the function f : R+ → R+ given by f (x) =
√
x is one-to-one and onto.
2. Prove that f is continuous at x0 = 0. (You may assume that x ≥ 0 for this so you
don’t leave the domain of the function.)
3. Prove that if x0 > 0 then f is continuous at x0 .
Problem 2. Consider the function g : R → R given by
(
x
x 6= 0
|x|
g(x) =
.
0 x=0
The function g is known as the signum function and is usually denoted sgn(x).
1. Prove that g is not continuous at x0 = 0.
2. Suppose that the value of g is changed at 0, keeping everything else about g the same.
Is it possible to do so in such a way that g becomes continuous? Why or why not?
(You don’t need to give a formal proof for this part, just a reasonable explanation.)
Problem 3–Extra Credit. Consider a clock which is normal in every way except that the
hour and minute hands are indistinguishable. Is it always possible to know what time it is?
Justify your response using the mathematics of continuous functions.
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