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2 Construct a function that is continuous at exactly four points. 3 Construct a...
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
Question 2 Please (2) (6 points) Indicate (by circling the letter) which of the following statements...
Question 2 Please
(2) (6 points) Indicate (by circling the letter) which of the following statements are true. (a) Every continuous function is differentiable. (b) Every differentiable function is continuous. (c) The function f(x) = 122 – 16 is differentiable at r = 4. (d) If f : [0,1] → R is a continuous function, then f has an absolute minimum. (e) If : [0,1] → R is a function, then / has an absolute maximum. () The product of...
this is Topology 3) Ifa functionf(R,T.)-(R,T) įs continuous, then f(R,Ts)-(R, т)is continuous. 4) If a function...
this is Topology
3) Ifa functionf(R,T.)-(R,T) įs continuous, then f(R,Ts)-(R, т)is continuous. 4) If a function EIR, ті )-(R,%) is continuous, then e (R, T)-cR,n) is continuous. 5) If a function f: (R )-(R, Tİ) İS continuous, then f(R,7, ) → (RM) t8 continuous. 6) If a function f:(R雨)-(RM) is continuous, then f (RM )-(RM) is continuous. 7) Any two discrete topological spaces are homeomorphic. 8) Any one-to-one, onto function between two discrete topological spaces is a homeomorphism
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3...
PLEASE ANSWER ALL! SHOWS STEPS
2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
2) Show that there is no continuous function f: RR that takes each actual valuve exactly...
2) Show that there is no continuous function f: RR that takes each actual valuve exactly twice that is, there is no function f such that for every CER there are exactly two real numbers a and b such that fca) = FCb) = C.
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typica...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
5. The inverse of a continuous invertible function in general is not continuous. But having a...
5. The inverse of a continuous invertible function in general is not continuous. But having a compact domain changes this: Suppose that K, A C R are sets and f: K + A is a continuous function with range A, that is, f(K) = A. Suppose also that K is compact and is invertible. Prove that the inverse function f-1: A + K is also continuous. Suggestion: Verify the sequential criterion for continuity. You may want to use the fact,...
3. Suppose that f [0,1(0,1) is a non-decreasing function (NOT assumed to be continuous). Prove or...
3. Suppose that f [0,1(0,1) is a non-decreasing function (NOT assumed to be continuous). Prove or disprove that there exists x E (0,1) such that f(x)-x
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. The function f is continuous on the closed interval (-2, 1). Some values of f...
4. The function f is continuous on the closed interval (-2, 1). Some values of f are shown in the table below. --2 f(x) -3 -1 0 1 7 k3 The equation f(x) = 3 must have at least two solutions in the interval [-1,1) if k = a. 1 b. C. 2 CONN NICO d. 5. If k(r) is a continuous function over the interval (-2, 4) such that k(-2) = 3 and k(4) = 1, then k(2) 0...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
Question 2 Please
(2) (6 points) Indicate (by circling the letter) which of the following statements are true. (a) Every continuous function is differentiable. (b) Every differentiable function is continuous. (c) The function f(x) = 122 – 16 is differentiable at r = 4. (d) If f : [0,1] → R is a continuous function, then f has an absolute minimum. (e) If : [0,1] → R is a function, then / has an absolute maximum. () The product of...
this is Topology
3) Ifa functionf(R,T.)-(R,T) įs continuous, then f(R,Ts)-(R, т)is continuous. 4) If a function EIR, ті )-(R,%) is continuous, then e (R, T)-cR,n) is continuous. 5) If a function f: (R )-(R, Tİ) İS continuous, then f(R,7, ) → (RM) t8 continuous. 6) If a function f:(R雨)-(RM) is continuous, then f (RM )-(RM) is continuous. 7) Any two discrete topological spaces are homeomorphic. 8) Any one-to-one, onto function between two discrete topological spaces is a homeomorphism
PLEASE ANSWER ALL! SHOWS STEPS
2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
2) Show that there is no continuous function f: RR that takes each actual valuve exactly twice that is, there is no function f such that for every CER there are exactly two real numbers a and b such that fca) = FCb) = C.
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
5. The inverse of a continuous invertible function in general is not continuous. But having a compact domain changes this: Suppose that K, A C R are sets and f: K + A is a continuous function with range A, that is, f(K) = A. Suppose also that K is compact and is invertible. Prove that the inverse function f-1: A + K is also continuous. Suggestion: Verify the sequential criterion for continuity. You may want to use the fact,...
3. Suppose that f [0,1(0,1) is a non-decreasing function (NOT assumed to be continuous). Prove or disprove that there exists x E (0,1) such that f(x)-x
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. The function f is continuous on the closed interval (-2, 1). Some values of f are shown in the table below. --2 f(x) -3 -1 0 1 7 k3 The equation f(x) = 3 must have at least two solutions in the interval [-1,1) if k = a. 1 b. C. 2 CONN NICO d. 5. If k(r) is a continuous function over the interval (-2, 4) such that k(-2) = 3 and k(4) = 1, then k(2) 0...
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