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.
Homework Answers
Consider the function f:R + R defined by if x is rational f(x) = if x...
Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C
1. Let f:R → R be the function defined as: 32 0 if x is rational...
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. The function f(x) = is not defined when r = -1, but it has a...
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.
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...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y)...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) =...
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
9. [7 points) Consider the function f(x) defined by f(x) = xeAs + B if x...
9. [7 points) Consider the function f(x) defined by f(x) = xeAs + B if x <3 C(x - 3)2 if 3 < x < 5 130 if > 5. C Suppose f(x) satisfies all of the following: f(x) is continuous at x = 3. • lim f(x) = 2 + lim f(x). 3+5+ 3-5- lim f(x) = -4. Find the values of A, B, and C. . 24-O
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a<...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks
3. Consider the function defined by...
1. Consider the function defined by (1 -2, 0 r< 1, f(x) 1 < |x2 (0. and f(r) f(x+ 4) (a) Sketch the graph of f(x...
1. Consider the function defined by (1 -2, 0 r< 1, f(x) 1 < |x2 (0. and f(r) f(x+ 4) (a) Sketch the graph of f(x) on the interval -6,61 (b) Find the Fourier seriess representation of f(x). You must show how to evaluate any integrals that are needed.
1. Consider the function defined by (1 -2, 0 r
Real analysis 2. Consider the following three definitions: A function f : R-+R is lax-continuous at a E R provided for...
Real analysis
2. Consider the following three definitions: A function f : R-+R is lax-continuous at a E R provided for all e > 0 there is a 6 > 0 such that for all r E R, if x - al6 then |f(x)- f (a)e A function f : R - R is e-continuous at a E R provided for all e >0 there is a 6 > 0 such that for all r E R, if |a- a...
Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C
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. 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.
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...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
9. [7 points) Consider the function f(x) defined by f(x) = xeAs + B if x <3 C(x - 3)2 if 3 < x < 5 130 if > 5. C Suppose f(x) satisfies all of the following: f(x) is continuous at x = 3. • lim f(x) = 2 + lim f(x). 3+5+ 3-5- lim f(x) = -4. Find the values of A, B, and C. . 24-O
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks
3. Consider the function defined by...
1. Consider the function defined by (1 -2, 0 r< 1, f(x) 1 < |x2 (0. and f(r) f(x+ 4) (a) Sketch the graph of f(x) on the interval -6,61 (b) Find the Fourier seriess representation of f(x). You must show how to evaluate any integrals that are needed.
1. Consider the function defined by (1 -2, 0 r
Real analysis
2. Consider the following three definitions: A function f : R-+R is lax-continuous at a E R provided for all e > 0 there is a 6 > 0 such that for all r E R, if x - al6 then |f(x)- f (a)e A function f : R - R is e-continuous at a E R provided for all e >0 there is a 6 > 0 such that for all r E R, if |a- a...
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