Description
1. (10 pts) (i) Where is the function f : R −→ R, given by
0 x is rational f(x) = 1 x is irrational,
continuous?
(ii) Where is the function g : R −→ R, given by
x x is rational
g(x) = 2x x is irrational,
continuous?
2. (10 pts) If f : Rn −→ Rm is any function, then
f(P) = (f1(P), f2(P), . . . , fm(P)) = f1(P)eˆ1+f2(P)eˆ2+· · ·+fm(P)eˆm,
where f1 : Rn −→ R, f2 : Rn −→ R, . . . , fn : Rn −→ R are functions.
(i) Show that if f is continuous, then so are f1, f2, . . . , fm.
(ii) Show that if f1, f2, . . . , fm are continuous, then so is f.
3. (10 pts) Let f : R2 −→ R be the function given f(x, y) = |xy|.
(a) Show directly that f is differentiable at (0, 0).
(b) Show that the partial derivatives are not continuous in any neighbourhood of the origin.
4. (10 pts) Find a function f : R2 −→ R such that
∂f = 3×2
y2
−xy sin(xy)+cos(xy) and ∂f = 2×3
y−x2 sin(xy)+3y2
. ∂x ∂y
5. (10 pts) (2.3.21).
6. (10 pts) (2.3.25).
7. (10 pts) (2.3.30).
8. (10 pts) (2.3.31).
9. (10 pts) (2.3.33)
10. (10 pts) (2.3.51).
Just for fun: (i) Let a ≥ 0 be a real number and let x be a real
number. How does one define ax? You may use the fact that if
a1 ≤ a2 ≤ a3 . . . ,
1
� �
is an increasing sequence of numbers which are bounded from above
(that is, there is a real number M such that ai ≤ M), then the limit
limn→∞ an exists.
(ii) Let a ≥ 0 be a real number. Show that the function f : R −→ R
given by f(x) = ax is continuous.
(iii) Let a1, a2, . . . , an be non-negative real numbers and let x1, x2, . . . , xn
be non-negative real numbers whose sum is one. Show that
n n
xi ai xiai ≤ .
i=1 i=1
2
18.022 Calculus of Several Variables
