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1 EL CIUSCUJUDUCU UL . sin(1/2), 220 Problem 6: (10 pts) Let f:R → R be...
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...
2. Let f:R + R and g: R + R be functions both continuous at a...
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...
ULUM turu Problem 1. Assume that f:R R is continuous and satisfies f(x) - f(y) =...
ULUM turu Problem 1. Assume that f:R R is continuous and satisfies f(x) - f(y) = (x - y)?, for all x, y ER. Show that f is constant.
Problem 1: Determine whether the statement is true or false. If the statement is true, then...
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x)...
(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.)
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex if for all 2, Y ER" and le (0,1), \f (2) + (1 - 1)f(y) = f(2x + (1 - 1)y). Using this definition, show that (a) f(3) = wfi (2) is a convex function for x ER" whenever fi: R → R is a convex function and w > 0 (b) f(x) = f1(x) + f2(2) is a convex function for x ER"...
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find its...
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find its Tavlor series up to and inchading the degree 2 term (6 marks F give rise to an inner 2 (c). Referring to the function F in part (b) above, for which values of a does the matrix A (4 marks product on R2? Show how you obtained your answer.
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find...
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr...
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr = 2Ě (2n-1) 2d. (10 pts) Let 0 < < be given. Show that f given in 2b) is differentiable at each 1 € (5,27 - 8). Find f' (1). Hint: Use Problem 1 and the following formula In 2 (-1)"-1 Σ 7 n=1 2. (40 pts) Let fn: R → R be given by fn (x) = sin (nx) 3 ηε Ν. n2...
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 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...
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...
ULUM turu Problem 1. Assume that f:R R is continuous and satisfies f(x) - f(y) = (x - y)?, for all x, y ER. Show that f is constant.
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
(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.)
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex if for all 2, Y ER" and le (0,1), \f (2) + (1 - 1)f(y) = f(2x + (1 - 1)y). Using this definition, show that (a) f(3) = wfi (2) is a convex function for x ER" whenever fi: R → R is a convex function and w > 0 (b) f(x) = f1(x) + f2(2) is a convex function for x ER"...
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find its Tavlor series up to and inchading the degree 2 term (6 marks F give rise to an inner 2 (c). Referring to the function F in part (b) above, for which values of a does the matrix A (4 marks product on R2? Show how you obtained your answer.
(Problem continued) 9 (b). Let F:R ([2) R be given by F In(z +1). Find...
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr = 2Ě (2n-1) 2d. (10 pts) Let 0 < < be given. Show that f given in 2b) is differentiable at each 1 € (5,27 - 8). Find f' (1). Hint: Use Problem 1 and the following formula In 2 (-1)"-1 Σ 7 n=1 2. (40 pts) Let fn: R → R be given by fn (x) = sin (nx) 3 ηε Ν. n2...
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
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