Consider the function f:R + R defined by if x is rational f(x) = if x...
Consider the function f : R → R defined by f(x) = !x if x is rational −x if x is irrational. Find all c ∈ R at which f is continuous. Consider the function f :R → R defined by .. х if x is rational f(x) = -2 if x is irrational. Find all c ER at which f is continuous.
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
5. Define f:R + R by f(x) = x2 if x is rational, and f(x) = 0 if x is irrational. Show that f'(0) exists and is equal to zero.
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
(2) Consider the function f given by f:R R f(a)1 2 (a) Determine the domain D and range R of the function f. (b) Show that f is not one to one on D. (c) Let ç D be a subset of the domain of f such that for all x ? S, 0 and the function is one to one. Find such a set S. (d) For the set S given in Part (c), find f (x) (e) Determine...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
5. The function f(x) = is not defined when r = -1, but it has a removable discontinuity there. Find a function g which agrees with f for x -1 and is continuous for all real numbers. 6. Find the intervals of increase and decrease of the function () == +692 +9a.
1. Consider the function: f:R R defined by f()2z +5 cos(3z). (a) Compute the values f (0) and f(1). Does f have a root in the interval [0, 1]? (b) Apply the first three steps of the bisection method for f by hand. You can use a pocket calculator for computations. What is the absolute error for the solution at the end? (c) Use Matlab to approximate the root with an absolute error of e0.5-10-6. (d) Compute the values f(0)...
help me. 5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and...