Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not uniformly continuous.

Q: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

A:

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: 2. Show that f(r) = sin r is uniformly continuous on (0, 00). %3D

A: 2. We need to prove that sinx is uniformly continuous on 0,∞. Definition of uniformly continuous…

Q: 3. Find two real-valued functions f and g that are uniformly continuous on a set E, but such that…

A:

Q: 24. Show that f(x) = mx + b, where m,b e R, is uniformly continuous on R. %3D

A:

Q: Let ƒ(x, y) =(x2-y2)/(x2+y2)  for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that…

A:

Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

A:

Q: Prove that the function sin(x?) is not uniformly continuous on [0, ). However, it is uniformly…

A: A function f(x) is said to be uniformly continuous in a set S if and only if ∀ε>0  ∃δ=δ(ε)  such…

Q: Px (w) is a continuous function of w in (- ∞, 0) for any real value of ow, and in fact, it is…

A:

Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

A:

Q: If f and g are continuous integrable functions on [a, b] then Lo f (x) + g (x) dx = [" ƒ (x) dx + S"…

A: To prove ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx , All are definite integrals with upper limit a and lower…

Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

A:

Q: Let f: X -> Y and g: Y -> Z be uniformly continuous on X and Y, respectively.  Prove g ◦ f: X -> Y…

A:

Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: (a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0

A: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: Clarify that f f :(0,1) →R defined by f(x): not uniformly continuous. =-

A:

Q: Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true,…

A: Let f: E → R be uniformly continuous.  Now, we need to show that |f| is uniformly continuous.  Now,…

Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

A:

Q: 4. Let f : X → Y and g : Y → Z be uniformly continuous on X and Y, respectively. Prove gof: X → Z is…

A:

Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -> R is a Lipschitz continuous function defined on X . To show : f is…

Q: Is f(x) = x² uniformly continuous on (0,1)? Why or why not?

A:

Q: B) Let f:[-2,2]R. then show that the function xsin -) for x 0, f(x) = for x = 0, is uniformly…

A:

Q: a) Let F(x) = | x· f(t)dt. Compute F'(x). 0, b) Assume that f is continuous. Prove that f(t)dt ) du.

A:

Q: Using the e, & definition of uniform continuity: Prove that f(x) = i, is uniformly continuous on…

A:

Q: If f satisfies a Lipschitz condition on ICR, then f is uniformly continuous on I. a.

A: Since you have sent more than one question, because of our guidelines we can just answer one of…

Q: Suppose that f: [0, ) – R is continuous, and that f(x) – l as x → - x. Prove that f is uniformly…

A: See the attachment

Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

A: Given that f be a function defined on ℝ satisfying…

Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

Q: Prove that if f is continuous on [a, b] and F(x) = ſx ƒ (t)dt α then F is continuously…

A:

Q: Show that f(z) =xy+iy is everywhere continuous but is not analytic

A: Answer

Q: Give an example of a function f which is integrable on (0,1] such that f(z) >0 for all (a) IE (0,1]…

A:

Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

A:

Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

Q: Let f(x, y) = Scosy.sinx, x # 0 cosy, x = 0 Is f continuous at (0,0)? Is ƒ continuous everywhere?

A: We have to check whether the given function is continuous at x = 0 or not. Also, check whether it is…

Q: 크) Prove that va +b 0. Prove that f is uniformly continuous on [0, 0).

A:

Q: Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0)…

A: Let F be defined on RXR→R by F(x,y) =x2-y. Now we have to show that F is in class C' on a…

Q: Let (s n(x-y) |x|+ly\ = 0 f(x, y)={ |x|+|y]| (0, (x, y) = (0,0) Is f continuous at the origin? Why?

A: The continuity of the function can be checked by checking the limits of the function at the point.…

Q: Show that the function f(x) = csc(x) is not uniformly continuous on (0, pi/2]

A:

Q: Let f (x) be a continuous function on R-{0} such that a a f (x) dx exists for all a > 0.If g(x) = [?…

A:

Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

A: we are given that f(x)=1   if x∈Q∩[a,b]-1 if x∈(RQ)∩[a,b] To prove f is not Riemann-integrable  on…

Q: If f is integrable on [a,b], show by example that F (t) = | f (x) dx is continuous but need not to…

A: We need to show by an example that if f is integrable on [a, b], then F(t) (as defined in the…

Q: The function f : (,,0) → R given by f(x) = it is not uniformly continuous on (;, 0). 1 is continuous…

A:

Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

A:

Q: 2. Let f (x) = cos-. Show that f is NOT uniformly continuous on (0,1).

A:

Q: Let f be the function Jo if x +1 and f(x) = 42 if x = 1. Prove that f is integrable on [0, 2] and…

A: Given :f is  a function such that               fx=0      if x≠1 and42    if x=1Now, on the interval…

Q: If f : R + Ris continuous and periodic, prove that f is uniformly continuous.

A: Given f:R→R is a periodic function with period k that is f(x+k)=f(x) To prove: if f:R→R is…

Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

A:

Q: give an example where f is not derivable at [0, ∞), but is uniformly continuous

A: give an example where f is not derivable at [0, ∞), but is uniformly continuous

Q: f (x) = %3D x² +1

A: