82 3.4 Properties of Continuous Functions Exercises 3.4.1 Let f : D→Rbe continuous at c ∈Dand let γ ∈R. Suppose f (c) >γ. Prove that there exists δ >0 such that f (x) >γ for every x ∈(c−δ,c+δ)∩D. 3.4.2 Let f,gbe continuous functions on [a,b]. Suppose f (a) <g(a) and f (b) >g(b). Prove that there exists x0 ∈(a,b) such that f (x0) =g(x0). 3.4.3 Prove that the equation cosx =x has at least one solution in R. (Assume known that the function cosx is continuous.) 3.4.4 Prove that the equationx2−2=cos(x+1) has at least two real solutions. (Assume known that the function cosx is continuous.) 3.4.5 Let f : [a,b] →[a,b] be a continuous function. (a) Prove that the equation f (x) =x has a solution on [a,b]. (b) Suppose further that | f (x)−f (y)| <|x−y| for all x,y ∈[a,b],x̸ =y. Prove that the equation f (x) =x has a unique solution on[a,b]. 3.4.6 ▷Let f be a continuous function on [a,b] and x1,x2, . . . ,xn ∈[a,b]. Prove that there exists c ∈[a,b] with f (c) = f (x1)+f (x2)+· · ·+f (xn) n . 3.4.7 ▷Suppose f is a continuous function onRsuch that | f (x)| <|x| for all x̸ =0. (a) Prove that f (0) =0. (b) Given two positive numbers aandbwitha<b, prove that there exists ℓ ∈[0,1) such that | f (x)| ≤ℓ|x| for all x ∈[a,b]. 3.4.8 ▶Let f,g: [0,1] →[0,1] be continuous functions such that f (g(x)) =g(f (x)) for all x ∈[0,1]. Suppose further that f is monotone. Prove that there exists x0 ∈[0,1] such that f (x0) =g(x0) =x0. 3.4.9 Prove Corollary 3.4.6.
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