Let F be the co
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- Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .Let f: E →Rp be continuous and differentiable in E, where E is an open subset of Rn. The mean value theorem holds for f if ____________________________.
- Show that the functions(a) f(x)=1/(1+x2),x∈R, (b) g(x)=x/(1+x2),x∈R, are uniformly continuous, while the functions(c) f(x)=x2,x∈R, (d) g(x)=√∣x∣,x∈R, are not Through graphSuppose that E is a measurable set and that f(x)=d for all x∈E. That is, f is a constant function on E. Prove that f is measurable.Let a, b ∈ R with a < b. Show that a function f : (a, b) → R is uniformly continuous on (a, b) if and only if it can be extended to a continuous function f˜ on [a, b]
- Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true, justifyLet f be a continuous function defined on E. Is it true that f^-1(A) is always measurable if A is measurable?For each of the following probability density functions, compute P (μ - 2σ < X < μ+ 2σ). f (x) = 6x (1 - x), 0 < x < 1, zero elsewhere. f (x) = (1/2)X, x = 1, 2, 3, ..., zero elsewhere.