The Real And Complex Number Systems

lx fc f cx c fx.
(Exercise 2) Letf : R R be convex. If f is bounded above, then f is a constant
function.
Proof: Suppose that f is not constant, say fa fb, wherea b. Iffb fa, we
consider
fx fb fb fa
,wherex b
x b b a
which implies that as x b,
fb fa
fx x b fb as x
b a
And if fb fa, we consider
fx fa fb fa
x a ,wherex a
b a
which implies that as x a,
fb fa
fx x a fa as x .
b a
So, we obtain that f is not bouded above. So, f must be a constant function.
(Exercise 3) Note that e x is convex on R. Use this to show that A. P. G. P.
Proof: Sincee x e x 0onR, we know that e x is convex. So,
ex 1 ...e
x n
n ,wherex i R, i 1, 2, . . . , n.
So, let e x i y i 0, for i 1, 2, . . . , n. Then
e x 1 ...xn
n
Vector-Valued functions
y 1 y n 1/n y 1 ...y n
n .
5.30 If a vector valued function f is differentiable at c, prove that
f c lim 1 fc h fc.
h0 h
Conversely, if this limit exists, prove that f is differentiable at c.
Proof: Write f f 1 ,...,f n : S R R n , and let c be an interior point of S. Then if
f is differentiable at c, each f k is differentiable at c. Hence,
lim 1 fc h fc
h0 h
f
lim 1 c h f 1 c
,..., f nc h f n c
h0 h
h
f
lim 1 c h f 1 c f
,...,lim n c h f n c
h0 h
h0 h
f 1 c,...,f n c
f c.
Conversly, it is obvious by above.
Remark: We give a summary about this. Let f be a vector valued function defined on
S. Write f : S R n R m , c is a interior point.
f f 1 ,...,f n is differentiable at c each f k is differentiable at c, *
and