1. Quizz 8
Definition 1.1. Let S be a subset of R and f : S → R be a function. Suppose that a is a limit point of S. We say that f has a limit L at a if for any > 0, there exists δ = δa, > 0 such that |f (x) − L| < whenever 0 < |x − a| < with x ∈ S.
Definition 1.2. Let S be a subset of R and f : S → R be a function. We say that f is continuous at a ∈ S if if for any > 0, there exists δ = δa,> 0 such that |f (x) − f (a)| < whenever |x − a| < with x ∈ S. If f is continuous at every point of S, we say that f is continuous (on S).
(1) Let (an) be a(ny) sequence in S such that an6= a for all n ≥ 1 and lim
n→∞an= a. If f has a limit L at a, prove that
n→∞lim f (an) = L.
Remark. The same technique can be used to prove the following. Let f : S → R be a function continuous at a. Let (an) be a(ny) sequence in S convergent to a. Prove that
n→∞lim f (an) = f (a).
(2) Let S = R \ {0} and define f : S → R by f (x) = cos 1 x
. (a) Prove that 0 is an accumulation point of S.
(b) Prove or disprove that f has a limit at 0.
(3) Let f : [a, b] → R be a continuous function. Prove that f ([a, b]) is closed and bounded.
(4) Let f : [a, b] → [a, b] be a function such that |f (x) − f (y)| ≤ C|x − y| for any x, y ∈ [a, b] for some 0 < C < 1. Prove that f has a unique fixed point in [a, b], i.e.
there exists a unique x ∈ [a, b] so that f (x) = x. (Hint: let x0 ∈ [a, b] and define xn = f (xn−1) for any n ≥ 1. Prove that (xn) is convergent in [a, b]. Prove that the limit of (xn) is a fixed point of f.)
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