Problems and Solutions in - Mathematics - University of Hawaii

1.4 2000 November 17 1 REAL ANALYSIS
1.4 2000 November 17
Do as many problems as you can. Complete solutions to five problems would be considered a good performance.
1. (a) 20 State the inverse function theorem.
(b) Suppose L : R 3 → R 3 is an invertible linear map and that g : R 3 → R 3 has continuous first order partial
derivatives and satisfies g(x) ≤ Cx 2 for some constant C and all x ∈ R 3 . Here x denotes the usual
Euclidean norm on R 3 . Prove that f(x) = L(x) + g(x) is locally invertible near 0.
Solution:
(a) (Inverse function theorem (IFT) of calculus) 21
Let f : E → R n be a C 1 -mapping of an open set E ⊂ R n . Suppose that f ′ (a) is invertible for some a ∈ E and
that f(a) = b. Then,
(i) there exist open sets U and V in R n such that a ∈ U, b ∈ V , and f maps U bijectively onto V , and
(ii) if g is the inverse of f (which exists by (i)), defined on V by g(f(x)) = x, for x ∈ U, then g ∈ C 1 (V ).
(b) First note that L and g both have continuous first order partial derivatives; i.e., L, g ∈ C 1 (R 3 ). Therefore, the
derivative of f = L + g,
f ′
∂fi
(x) Jf (x)
∂xj
exists. Furthermore, Jf (x) is continuous in a neighborhood of the zero vector, because this is true of the partials of
g(x), and the partials of L(x) are the constant matrix L. Therefore, f ∈ C 1 (R 3 ). By the IFT, then, we need only
show that f ′ (0) is invertible. Since f ′ (x) = L + g ′ (x), we must show f ′ (0) = L + g ′ (0) is invertible. Consider
the matrix g ′ (0) = Jg(0). We claim, Jg(0) = 0. Indeed, if x1, x2, x3 are the elementary unit vectors (also known
as i, j, k), then the elements of Jg(0) are
∂gi
∂xj
3
i,j=1
gi(0 + hxj) − gi(0)
(0) = lim
= lim
h→0 h
h→0
gi(hxj)
. (8)
h
The second equality follows by the hypothesis that g is continuous and satisfies g(x) ≤ Cx 2 , which implies
that g(0) = 0. Finally, to show that (8) is zero, consider
which implies
|gi(hxj)| ≤ g(hxj) ≤ Chxj 2 = C|h| 2 ,
|gi(hxj)|
|h|
2 |hxj|
≤ C = C|h| → 0, as h → 0.
|h|
This proves that f ′ (0) = L, which is invertible by assumption, so the IFT implies that f(x) is locally invertible
near 0. ⊓⊔
2. Let f be a differentiable real valued function on the interval (0, 1), and suppose the derivative of f is bounded on
this interval. Prove the existence of the limit L = lim x→0 + f(x).
20 The inverse function theorem does not appear on the syllabus and, as far as I know, this is the only exam problem in which it has appeared. The
implicit function theorem does appear on the syllabus, but I have never encountered an exam problem that required it.
21 See Rudin [7].
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