HW9, Due November 7

HW9, Due November 7
1. Find the limits and justify your result using either the -δ or Sequential Criterion for limits
1
x→2 1 − x
lim [[x]] = 2,
lim
x→e
x2
x→0 |x|
where [[x]] denotes the greatest integer less than or equal to x
lim
2. Use the -δ definition of the limit to show that
lim (x2 + 4x) = 12
x→2
and
x+5
=4
x→−1 2x + 3
lim
3. Let c ∈ R and let f : R → R be such that limx→c (f (x))2 = L.
(a) Show that if L = 0, then limx→c f (x) = 0
(b) Show by example, that if L 6= 0, then f may not have a limit at c.
4. Prove that the following functions defined on R are not continuous
(a) The Dirichlet function from section 4.1 of Abott
(
x
sin( |x|
), x 6= 0
(b) The function f (x) =
sin 1, otherwise
What are the sets of discontinuities in the example above? (You may assume that the sine
function is continuous)
5. Prove the following theorem
Theorem 1. Let A ⊂ R and f : A → R. Let c ∈ R be the limit point of A. Then if
limx→c f (x) > 0, there exists a neighborhood Vδ (c) of c, such that for all x ∈ A ∩ Vδ (c) and
x 6= c, f (x) > 0.
6. Is the an analogue of Theorem 1 valid with limx→c f (x) ≥ 0? Prove or provide a counterexample.
7. Prove that the composition of continuous functions is continuous (Theorem 4.3.9 on p. 112
of Abott)
√
√
8. Prove that f (x) = n x is continuous. As with x the argument for x 6= 0 should be treated
separately (as same ε-δ argument will not apply to both). The proof is broken into the
following steps
√
(a) Review the proof for the continuity of x.
√
(b) Prove limx→0 n x = 0
P
P1
n−j−1 y j (e.g. x2 − y 2 = (x − y)
1−j y j )
(c) Verify that xn − y n = (x − y) · n−1
j=0 x
j=0 x
√
√
√
(d) Prove that limx→c n x = n c (you may work out 3 x case separately as a warm up)
9. Given a function h : R → R define h−1 (A) = {x ∈ R : h(x) ∈ A}. Prove that the set h−1 (0)
is closed for a continuous function h.
10. Generalize the previous example to a preimage of a closed set. That is if B ∈ R is closed,
then h−1 (B) is closed for a continuous h : R → R.
1
`