If f is a continuous function satisfying f(x) = x -x+(1-x),e sodi+xf (t-r)f@)dı for all x, then f "(1) = ? A) -1 B) 0 C) 1 D) 2 E) 3

Q: 3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing…

A:

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

A:

Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Show that if u =x - y and v =r+ y, then for any continuous f, e-2s* f(x - y,x + y)dydr = || (a+n)e-…

A:

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

A:

Q: Let f (x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that () = 4u du + ax 2.

A: Given, f(x,y)=g(u)=x2+y2 To prove that, ∂f∂x2+∂f∂y2=4udgdu2

Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: Z are continuous, then their composition Theorem 7.9. If f : X → Y and g gof : X → Z is continuous.…

A: The given theorem is as follows. If f:X→Y,g:Y→Z are continuous, then their composition g∘f:X→Z is…

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

A:

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

A:

Q: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

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

A:

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

A:

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

A:

Q: If f : a, b] → Ris differentiable at a point c E (a, b), then f is continuous at c. True False

A:

Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

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

A:

Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

A:

Q: As the fun ction Leay) xy js Continuous I+ x*+y? e (-1, 2)

A:

Q: (a) Assume that g(x, y) is a function of the form g(x, y) = {" 0 if (x, y) = (0,0). (i) Show that…

A:

Q: Prove that if f, g :R → R are continuous at x E R, then ƒ+gis continuous at x E R as well.

A: By using the definition of continuous function this above statement can be easily proved.

Q: Show that f(x) = In x is not uniformly continuous on (0,2)

A:

Q: 6. Let f: R? → R? be a continuous function. Prove that f ([0,1] × [0,1]) is compact.

A:

Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: (b). Show that iff= u+iv is analytic in a region S and u is a constant function (i.e., independent…

A:

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: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

A:

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

A:

Q: Let f : [0 , 1] → R be a continuous function on [0 , 1] such that : f(x)x" dx = 0, (n = 0, 1, 2,…

A: Given that f:0,1→R be a continuous function on 0,1 such that ∫01f(x)xndx=0. The objective is to show…

Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [0,…

A:

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

A:

Q: Let f, g : D → R be continuous at c E D. Prove that fg is continuous at c.

A: Let ε&lt;1 and M= maxf(a), g(a), 1, then there exists δ&gt;0 such that f(x)-f(a)&lt;ε3M (since f is…

Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

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: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

A: Recall definition of conituity as, Right hand limit = left hand limit = value of function at the…

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

A:

Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

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: The function f(x) interval [1, e] at x = Hora lo 0 x is the nat

A:

Q: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

A:

Q: If f is a continuous function and F'(x)=f(x) for all x € R, then - x)dx=?

A:

Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

A:

Q: 3. Let f, g : D →R be continuous at c € D. Prove that fg is continuous at c.

A: Since you have asked multiple questions, we have answered the first question. If you want the second…

Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

A: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A:

Q: Suppose that the function f: R → Ris continuous and that its image f (R) is bounded. Prove that…

A:

Q: If X is a continuous RV, prove that E(X) = [1- F(x)] dx – F(x) dx. %3D

A:

Q: Exercise 11. Prove that if f: E R is continuous at p E E, then there exists & >0 so that f(x) is…

A: