82 3.4 Properties of Continuous Functions Exercises 3.4.1 Let f : D → R be continuous at c ∈ D and 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,g be 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 equation x2 − 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],x ̸ = 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 (x1)+ f (x2)+ ··· + f (xn) f (c)= . n 3.4.7 ▷ Suppose f is a continuous function on R such that | f (x)| < |x| for all x ̸ = 0. (a) Prove that f (0)= 0. (b) Given two positive numbers a and b with a < 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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