M2PM1 Sheet 1
1. Let f : R3 \{0} → R be defined by f (x) = 3(x21 + x23 ) + 1 for x 6= 0. Prove, directly from
the definition of a limit, that f (x) → 1 as x → 0.
x1 + 2x2
2. Show that function defined by f (x1 , x2 ) = p 2
for (xi , x2 ) ∈ R2 \{(0, 0)} does not
2
x1 + 3x2
have a limit as x → 0.
3. Suppose that f : R → R is continuous at a and that f (a) 6= 0. Prove that there is a δ > 0
such that |x − a| < δ implies |f (x)| > 12 |f (a)|. Can you prove that such δ exists if f (a) = 0 ?
[please give a proof or a counterexample here]
4. Let f : Rn → R be continuous at a ∈ Rn . Prove that there is some δ > 0 such
that f is bounded on the ball Bδ (a). Moreover, show that δ > 0 can be chosen so that
|f (x)| ≤ |f (a)| + 1 for all x ∈ Bδ (a).
5. Let f : R → R be such that f (x) → 1 as x → 0. Prove that for every n ∈ N there is some
m ∈ N such that f (1/m) < 1 + 1/n.
6. Let f be continuous at a. Under what conditions is 1/f 2 continuous at a? Give reasons
for your answer.
7. For some α, β ∈ R define f (x) = αx for x ∈ Q and f (x) = βx for x 6∈ Q.
(i) If α 6= β, find all points a ∈ R such that f is continuous at a.
(ii) Find all values of α, β such that f 2 is continuous at all points.
(iii) Find all α, β ∈ Q\{0} such that f ◦ f is continuous at all points.
8. Prove that at any given moment there are two antipodal points on the equator at which
the temperature is the same. (Hint: let f (x) be the temperature at the point on the equator
with longitude x, and consider the function f (x + π) − f (x) on the interval [0, π]. You should
assume that f is continuous on [0, 2π].)
9. Let f be continuous on R such that f (x) is rational for all x. Prove that f is constant.
10. In each of the following cases, decide whether the given function (a) is continuous; (b)
is bounded; (c) attains its bounds on the given interval. If the interval is closed, does the
function take every value between its values at the end-points?
(i) 1 − x2 on (−1, 1); (ii) x − [x] on [−1, 1]; (iii) 1 − x + [x] − [1 − x] on [0, 1].
Here [x] denotes the “integer part” of x.
11. Let f be continuous on [a, b]. Show that there exists c ∈ [a, b] such that
1
f (c)2 = (f (a)2 + f (b)2 ).
2
1
M2PM1 Sheet 2
1. Let f be the function defined by f (x) = x2 sin(1/x) for x 6= 0 and f (0) = 0. Prove that
f is differentiable at 0 and find f 0 (0).
2. Let g be a function defined for all x in some open interval containing a, and let f be
defined by f (x) = (x − a)g(x).
(i) Prove that if g is differentiable at a, then f is too, and that in this case f 0 (a) = g(a).
(ii) Is it possible for f to be differentiable at a even if g is not? And if so, is it still necessarily
true that f 0 (a) = g(a)? And if not, when is f 0 (a) = g(a)?
3. Let f be the function defined in Q1, and let g be a function such that g 0 (x) = sin(sin(x+1))
and g(0) = 2. Find (f ◦ g)0 (0) and (g ◦ f )0 (0).
4. Prove that if f is differentiable at a, then
f (a + h) − f (a − h)
.
h→0
2h
f 0 (a) = lim
Give an example where this last limit exists, but f is not differentiable at a.
5. Suppose f is differentiable everywhere on R. Prove that if f is even then f 0 is odd.
6. Prove (directly from the definition) that the function f (x) = x12345 is differentiable at 0
and calculate f 0 (0).
7. Let f (x) = cx , where c > 0. Prove that f is differentiable at every point a ∈ R, and find
f 0 (a).
8. Let f (x) = xc , where c > 0. Prove that f is differentiable at every point a > 0, and find
f 0 (a).
9. Let f (x) = xx . Prove that f is differentiable at every point a > 0, and find f 0 (a).
x
10. Let f (x) = xc , where c > 0. Prove that f is differentiable at every point a > 0, and
x
x
find f 0 (a). As usual xc means x(c ) .
11. Let f (x) = cx , where c > 0 and c 6= 1. Find the inverse function of f and compute its
derivative.
1
M2PM1 Sheet 3
1. Prove the following fact: suppose that we have f (x) → l as x → a, and that f (x) ≥ C
for all x close enough to a (where C is a constant). Then l ≥ C.
2. Suppose that f is continuous on [a, b] and differentiable on (a, b). Prove that the following
two statements are equivalent:
(i) f 0 (x) ≥ 0 for all x ∈ (a, b);
(ii) if x1 and x2 are points of [a, b] such that x1 ≥ x2 , then f (x1 ) ≥ f (x2 ).
Give an example to show that this assertion would be false if we replaced “≥” by “>”
everywhere.
3. Let f (x) = |x|, a = −1, b = 2. Prove that the statement
there is some c ∈ (a, b) such that f 0 (c) =
f (b) − f (a)
b−a
is false. Why does this not contradict the Mean Value Theorem?
4. Prove that | sin x1 − sin x2 | ≤ |x1 − x2 | for all x1 and x2 . You may assume here that all
the things you have heard about functions sin x and cos x are true.
5. Prove that if f is a function on an interval [a, b] satisfying
|f (x1 ) − f (x2 )| ≤ (x1 − x2 )2 for all x1 , x2 ∈ [a, b],
then f is constant on [a, b].
6. Prove that the polynomial x3 −3x+β never has two roots in [−1, 1], whatever the constant
β is.
7. Suppose f is differentiable everywhere on R. Prove that if f 0 is odd then f is even. What
can you say about f if f 0 is even?
1
M2PM1 Sheet 4
1. Prove the following (frequently used) generalisation of L’Hôpital’s rule. Suppose that for
some n ≥ 1, f (r) (x) and g (r) (x) approach 0 as x → a for all r such that 0 ≤ r < n; and that
f (n) (x)/g (n) (x) → l as x → a. Then f (x)/g(x) → l as x → a.
Hint: use induction, starting from the case n = 1 proved in the lectures.
2. What is wrong with the following use of L’Hôpital’s rule?
x3 + x − 2
3x2 + 1
6x
=
lim
= 3.
=
lim
x→1 x2 − 3x + 2
x→1 2x − 3
x→1 2
lim
3. Let f (x) = 1/(1 − x). For any n ≥ 0 find f (n) (x), then f (n) (0), and hence find the
Taylor series of f at 0. Without using Taylor’s theorem, prove that the series converges to
f provided that |x| < 1.
4. Let
x2
x2 x 3
+ R3 = x −
+
+ R4 .
2
2
3
Use the Lagrange form of the remainder in Taylor’s theorem to find explicit expressions for
R3 and R4 , and hence prove that
log(1 + x) = x −
x2
x2 x3
< log(1 + x) < x −
+
for all x > 0.
2
2
3
What happens to these inequalities if −1 < x < 0?
x−
5. Let f be a function differentiable any number of times everywhere in R, and such that
(i) f 00 (x) = f (x); (ii) f (0) = 0; (iii) f 0 (0) = 1. Find the Taylor series of f at 0. Using
Lagrange’s form of the remainder, prove that the series converges to f (x) for all x ∈ R.
6. Let f be continuous on [0, 1] and differentiable on (0, 1). Prove that there is a point
0
c ∈ (0, 1) such that f (1) − f (0) = f 2c(c) .
Hint: Cauchy’s MVT might be useful.
7. Is it true that if f is differentiable any number of times at a and all of its derivatives are
continuous at a, then f is equal to its Taylor’s series at a?
8. Prove that if f (2009) (a) = 0 for every a ∈ R, then f (x) is a polynomial. Here, as usual,
f (2009) (a) means the 2009th derivative of f at a.
1
M2PM1 Sheet 5
1. Let f be a function differentiable any number of times on some open interval containing
a, and let
n
X
f (r) (a) r
Pn (h) =
h
r!
r=0
be the nth Taylor polynomial of f at a. Prove that for any n ≥ 1,
f (a + h) − Pn (h)
→ 0 as h → 0 ;
hn
and that Pn is the only polynomial of degree ≤ n that has this property.
2. Let f be a function that is differentiable on some open interval containing a, and such
that f 00 (a) exists. Prove that
f (a + h) + f (a − h) − 2f (a)
→ f 00 (a) as h → 0.
2
h
Hint: use L’Hôpital’s rule and one of the questions from earlier problem sheets.
3. Let f be the function on [−1, 1] defined as follows: f (x) = 0 if x 6= 0, and f (0) = 2.
(i) Prove that all the lower sums s(f, Δ) are zero, and hence find the lower integral j(f ).
(ii) Prove that all the upper sums S(f, Δ) are positive.
(iii) Given any > 0, find a partition Δ of [−1, 1] such that S(f, Δ ) < , and hence find
the upper integral J(f ).
(iv) Deduce that f is Riemann integrable over [−1, 1], and find its integral.
4. Let f be the function f (x) = x on [0, 1], and let Δn be the partition of [0, 1] into n
subintervals of equal length. Find explicit formulae for the lower and upper sums s(f, Δn )
and S(f, Δn ). Deduce that f is Riemann integrable over [0, 1], and find its integral.
5. Let f : [0, 2] → R be right continuous at 0 and left continuous at 2. Assume that f has
two derivatives f 0 , f 00 on (0, 2).
(i) Prove that f is continuous on the closed interval [0, 2].
(ii) Suppose further that f (0) = f (1) = f (2) = 0. Prove that there is a point c ∈ (0, 2) such
that f 00 (c) = 0.
(iii) Given in addition that |f 0 (x)| ≤ 1 for all x ∈ (0, 2), prove that |f (x) − f (y)| ≤ |x − y|
for all x, y ∈ [0, 2].
1
M2PM1 Sheet 6
1. Prove that if f is Riemann integrable over [a, b], then for any > 0 there is a partition Δ
of [a, b] such that S(f, Δ) − s(f, Δ) < .
R2. Prove that if f is Riemann integrable over [a, b] and f (x) ≥ 0 for all x ∈ [a, b], then
f ≥ 0. Deduce that
R if fR and g are both Riemann integrable over [a, b] and f (x) ≥ g(x) for
all x ∈ [a, b], then f ≥ g.
3. If f is a bounded function on [a, b], define f+ (x) = max{f (x), 0} and f− (x) = max{−f (x), 0},
so that f = f+ − f− and |f | = f+ + f− . Prove that if f is Riemann integrable over [a, b],
then so are f+ and f− . [Hint: use Theorem 3.6 and Q1]. Deduce that |f | is also Riemann
integrable over [a, b].
4. Prove that |
R
f| ≤
R
|f |.
5. Give an example in which J(f + g) 6= J(f ) + J(g).
6. Prove that if f is a bounded function on [a, b] and (Δn ) is any sequence of partitions of
[a, b] such that S(f, Δn ) − s(f, Δn ) → 0 as n → ∞, then f is Riemann
integrable over [a, b],
R
and both the sequences (S(f, Δn )) and (s(f, Δn )) converge to f .
7. Let f (x) = 1/(1 + x2 ), and let Δn be the partition of [0, 1] into n equal parts. Prove that
S(f, Δn ) − s(f, Δn ) → 0 as n → ∞. Hence show that
!
n
X
1
π
= .
lim n
2
2
n→∞
n +r
4
r=1
8. Let f, g : [a, b] → R be two bounded functions. Prove (directly from definitions) that
J(f + g) ≤ S(f, Δ) + S(g, Δ) for all partitions Δ of [a, b].
1
M2PM1 Sheet 7
1. Let f (x, y) = x2xy
for (x, y) 6= (0, 0), and f (0, 0) = 0. Prove that this function is
+y 2
2
bounded on R .
Find an example of a function f : R2 → R such that (i) for any fixed y ∈ R the function
x 7→ f (x, y) is differentiable on R; (ii) for any fixed x ∈ R the function y 7→ f (x, y) is
differentiable on R; (iii) f is not bounded on any ball centred at (0, 0).
So, the existence of partial derivatives does not even imply that the function is bounded
on arbitrarily small sets.
2. Prove that if ∇f (x) = 0 for all x ∈ Rn , then f is a constant function.
3. Let ν ∈ Rn , and let functions f, g : Rn → R be defined by f (x) = x ∙ ν and g(x) = |x|2 .
Let a ∈ Rn be some point. Show that f and g are differentiable at a and find ∇f (a) and
∇g(a), using the definition of the gradient.
4. For x, y ∈ R, let us define functions px , qy : R → R2 by px (y) = (x, y) and qy (x) = (x, y).
Prove that these functions are continuous everywhere.
Let now f : R2 → R be differentiable at all points. Let f1 , f2 : R → R be defined by
f1 = f ◦ px and f2 = f ◦ qy . Argue that f1 , f2 are differentiable at all points and show that
f10 =
∂f
,
∂y
f20 =
∂f
.
∂x
For each x, y ∈ R, let us finally define g1 (x) = f ◦ px (y) and g2 (y) = f ◦ qy (x). Argue that
g1 , g2 are differentiable at all points and show that
g10 =
∂f
,
∂x
g20 =
∂f
.
∂y
Hint: this exercise is much simpler than it may seem.
5. Let ν ∈ Rn be such that |ν| = λ. Let f : Rn → R be differentiable al all points. Prove
that
∂f ≤ λ|∇f |.
∂ν 1
M2PM1 Sheet 8
1. Prove the validity of the following statements:
a. The intervals (a, b), (a, ∞), (−∞, b) are open sets for any a, b ∈ R.
b. The intervals [a, b], [a, ∞), (−∞, b] are closed sets for any a, b ∈ R.
c. The intervals [a, b), (a, b] are neither closed, nor open sets for any a, b ∈ R.
d. Any finite set A = {a1 , .., an } ⊂ R is a closed set.
e. The intervals (a, b), (a, ∞), (−∞, b), [a, ∞), (−∞, b] are not compact sets.
f. Any finite set A = {a1 , .., an } ⊂ R is a compact set.
g. The set A = { n1 , n > 0} is not a compact set.
2. Prove de Morgan’s laws, that is prove that
!c
\
[
Eα =
Eαc
α∈J
\
α∈J
Eα
α∈J
!c
=
[
Eαc
α∈J
3. Let F be the Cantor set. The Cantor set is the set constructed as follows: Let F0 be
1 2
, 3 from F0 and denote the resulting set by F1 . Then
interval [0, 1]. We remove the interval
3 1 2
7 8
we remove the intervals 9 , 9 and 9 , 9 from F1 and denote the resulting set (consisting
of four intervals) by F2 . From each of these four intervals we remove the middle interval of
length 313 , and so forth. If we continue this process we obtain a decreasing sequence of sets
Fn . Define
\
F =
Fn .
n∈N
Prove that the Cantor set is a closed set.
4. Let f : R → R be a function continuous everywhere, A ∈ R and let
f −1 (A) = {x ∈ R|f (x) ∈ A}.
a.
b.
c.
d.
e.
If A is an open set, prove that f −1 (A) is open.
If A is a closed set, prove that f −1 (A) is closed.
If A is a compact set, does it follow that f −1 (A) is compact ?
If A is a finite, does it follow that f −1 (A) is finite ?
Can f (R) be a finite set ?
5. Prove that the following sets are Borel sets:
a. The intervals (a, b), (a, ∞), (−∞, b), [a, b], [a, ∞), (−∞, b].
b. Any finite or countable set A ∈ R.
c. Any open or closed set.
d. The Cantor set.
6. Prove that any continuous function is Borel measurable.
7*. Are there sets A ∈ R that are not Borel sets?
1
M2PM1 Sheet 9
1. Prove that the function f : R2 → R defined as
(
√ x2 2 for (x, y) 6= (0, 0)
x +y
f (x, y) =
0
for (x, y) = (0, 0)
is bounded. Find the partial derivatives for the function f for any (x, y) ∈ R2 , (x, y) 6= (0, 0).
Does f have partial derivatives at (0, 0) ?
2. Find the partial derivatives for the function g : R2 → R defined as
(
√ xy
for (x, y) 6= (0, 0)
x2 +y 2
g (x, y) =
0
for (x, y) = (0, 0)
Is g differentiable ? Is g continuous ?
3. Find the partial derivatives for the function h : Rn → R defined as h (x) = |x| , i.e.
q
h (x1 , x2 ...xn ) = x21 + ... + x2n .
Does h have partial derivatives at (0, 0) ? Is h continuous?
4. Prove that the function i : Rn → R defined as
i (x) = h (x)3 ,
where h is the function defined in Question 3, is differentiable everywhere. Compute the
gradient of the function.
5. Let j : R2 → R be the function defined as
( 2 2
j (x, y) =
x y
x2 +y 4
for (x, y) 6= (0, 0)
for (x, y) = (0, 0)
0
i. Find the directional derivative of j at (0, 0) in the direction (a, 1)
ii. Find the directional derivative of j at (0, 0) in the direction (ma, m). What do you
observe ?
6. Let k : R2 → R be the function defined as
(
2
k (x, y) =
xy
x2 +y 4
0
f or (x, y) 6= (0, 0)
f or (x, y) = (0, 0)
i. Prove that k has directional derivatives at (0, 0) in any direction.
ii. Prove that k is not continuous at (0, 0) ?
1