1. hw 8 (1) Show that cos x = x3 has at least one real root.
(2) Construct a function with discontinuities satisfying the intermidiate value property.
(3) Find a real-valued function f on an interval I of R such that infIf exists but minIf does not exist.
(4) Let f (x) be an increasing function on [0, 1] satisfying the intermediate value property. Show that f (x) ∈ C[0, 1].
(5) Suppose f (x), g(x) ∈ C[0, 1]. f (0) < g(0) and f (1) > g(1). Show that there exists x0∈ (0, 1) so that f (x0) = g(x0).
(6) Let f (x), g(x) ∈ C[0, 1]. Suppose f (x) 6= 0 for all x ∈ [0, 1]. If (f (x))2 = (g(x))2 for all x ∈ [0, 1], show that either f (x) = g(x) for all x ∈ [0, 1] or f (x) = −g(x) for all x ∈ [0, 1].
(7) Let f (x) = sin x, x ∈ R.
(a) Show that | sin x1− sin x2| ≤ |x1− x2| for any x1, x2∈ R.
(b) Show that f is uniformly continuous.
(8) Suppose f : R → R is a function such that there exists C > 0 so that
|f (x) − f (y)| ≤ C|x − y|
for any x, y ∈ R. (14) is an example of such a function.
(a) Show that f is uniformly continuous on R.
(b) Assume further that C < 1. Let x1be a point in [a, b]. Define a sequence of real numbers (xn) by
xn+1= f (xn), n ≥ 1.
Show that (xn) is convergent.
(c) As above, denote the limit of (xn) by x. Show that f (x) = x.
(d) As above, prove that the equation f (x) = x has a unique solution.
(9) Construct a continuous function on R which is not uniformly continuous on R.
(10) Let f : (a, b) → R be a uniformly continuous function. In this exercise, we are going to show that f has a unique extension to a continuous function F : [a, b] → R, i.e. F : [a, b] → R is a continuous function such that F |[a,b]= f.
(a) Let (xn) be a Cauchy sequence in (a, b), i.e. (xn) is a sequence of real numbers such that a < xn < b for all n ≥ 1. Show that (f (xn)) is also a Cauchy sequence.
(b) Suppose that (xn) and (yn) are both sequences in (a, b) convergent to a. Show that the limits of (f (xn)) and (f (yn)) exist and lim
n→∞f (xn) = lim
n→∞f (yn). Similarly, if (zn) and (wn) are both Cauchy sequences in (a, b) convergent to b, we also have lim
n→∞f (zn) =
n→∞lim f (wn).
(c) Define a new function F : [a, b] → R by
F (x) =
f (x) if a < x < b, limn→∞f (xn) if x = a, limn→∞f (zn) if x = b.
Here lim
n→∞xn= a and lim
n→∞zn= b. Show that F : [a, b] → R is (uniformly) continuous.
(d) Let G : [a, b] → R be a continuous function such that G|(a,b)= f. Show that G = F.
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