Problems 7
Please, do problems 3, 8, 12bdefh, 14, 15, 16, 17, 18.
1. Show that
0
x=0
1
6 0
sin x x =
f (x) =
is not continuous at 0.
2. Show that
0
x=0
1
6 0
x sin x x =
x x∈Q
0 x∈R\Q
f (x) =
is continuous at 0.
3. Show that
f (x) =
is continuous at 0, but nowhere else.
4. Give an example of a function f , such that f is nowhere continuous, but |f | is continuous every
where.
5. Suppose that f is a function satisfying |f (x)| ≤ |x| for all x. Show that f is continuous at 0.
6. Show that
f (x) =
0
1
q
x irrational
x = pq in lowest terms
is continuous for irrational x.
7. Suppose f (x+y) = f (x)+f (y) for all x, y and that f is continuous at 0. Show that f is continuous
at a for all a.
8. Prove: Suppose that f is continuous at l and limx→a g(x) = l, then limx→a f (g(x)) = f (l).
Note: g need not be continuous or even defined at a.
9. Let f be a function whose domain is the entire real line. If A and B are disjoint sets, does it
follow that f (A) and f (B) are disjoint sets?
Proof or counterexample.
If C and D are disjoint sets, does it follow that f −1 (C) and f −1 (D) are disjoint sets?
Proof or counterexample.
10. Let f be a function whose domain is the entire real line. If A and B are sets, does it follow that
f (A ∪ B) = f (A) ∪ f (B)?
Proof or counterexample.
If C and D are sets, does it follow that f −1 (C ∪ D) = f −1 (C) ∪ f −1 (D)?
Proof or counterexample.
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11. Give an example of two functions discontinuous at x = 0, whose sum is continuous at x = 0.
Give an example of two functions discontinuous at x = 0, whose product is continuous at x = 0.
12. Definitions:
Let I be an interval and f : I → R, c ∈ I. f is discontinuous at c, if f is not continuous at c.
Four types of discontinuities:
(i) f has an infinite discontinuity at c if either limx→c+ f (x) or limx→c− f (x) is infinite.
(ii) f has a jump discontinuity at c if either limx→c+ f (x) or limx→c− f (x) are finite but
unequal.
(iii) f has a removable discontinuity at c if limx→c f (x) exists and is finite, but f (c) is
not defined to have the value of limx→c f (x).
(iv) f has an oscillating discontinuity at c if the function f is bounded and the one-sided
limits do not exist.
Find all the discontinuities for each of the following functions, and identify each one as removable,
infinite, jump or oscillating.
(a) f (x) =
x2 +2x
x2 −4
(b) f (x) = [x + 12 ]
(c) f (x) = [ x1 ]
(d) f (x) =
sin x
x
(e) f (x) = x sin x1
√
(f) f (x) =
x−2
x−4
(g) f 0 (x), where f (x) = |x|
(h) f (x) = 0, where x ix rational and f (x) = 1, where x is irrational
(i) f (x) =
x2 +2x
x3 −x2
13. Show that the discontinuity of f (x) = x1 does not fit any of the four categories above. How might
the categories be redefined so as to handle this type of discontinuity?
P
14. Prove that if the functions f1 , f2 , ..., fn are all continuous at c ∈ I, then the function f = ni=1 fi
will also be continuous at c ∈ I.
p
15. Show that f (x) = |x| is continuous at x = c.
16. Show that f (x) =
1
x+2
is continuous at x 6= −2.
17. Show that f (x) = x3 is continuous at x = c.
18. If f, g are continuous functions defined on some subset D ⊂ R prove that max(f, g) and min(f, g)
are continuous functions, where max(f, g)(x) = max(f (x), g(x)) and min(f, g)(x) = min(f (x), g(x)).
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