IF f:x-у and vCy proof fy/v)=x/f"(v) and proof f is continous iff f'(E) is closed set in x for every closed set E in y

Q: Let f(x+y)-(x)(y) for all x, y e R and suppose that fis differentiable at x = 0.Show that fis…

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Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: Prove that f(z)= 1/(1-z) in not uniformly continuous for |z|

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Q: Show

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Q: Theorem : Let f be defined on [a, b] , if f has a local minimum at a point x € (a, b), and if f'(x)…

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Q: Prove that CL dx = ln lx1+C. Chint): set x

A: Please refer to the image below.

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

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Q: ɔ that h is a continuous functi (x2 — 2, х < 6 h(x) 3= 7х + a, х 6

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Q: Suppose that f is a function on [a, b] such that f(x) = 0 for all x e [a, b]\T , where T is finite…

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Q: [0,1] 6. If f is continuous on • prove that

A: We can solve an integral of some function f(x) w.r.t. x by using substituting, say, t=f(x) which…

Q: Let 1-tan r f(x)= ExtIE 0,5], 1f f(x) is continuous in [0.), then is 1 -1 /2

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Q: 2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem…

A: Given: a) f(x, y) =x - y - x2 Theorem 2.3.1: This theorem says that if f is twice continuously…

Q: Show that f(x) = 1 x is not uniformly continuous on (0,1) but it is uniformly continuous on [1,2]

A: The function is continuous at x=a if, lim(x→a+)= lim(x→a-)=f(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 -&gt; R is a Lipschitz continuous function defined on X . To show : f is…

Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: 2. If X is a finite set and fn → f on X, show that fn = f on X. 3. If fn 3 f on X and Y is a…

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Q: 8. Prove that a function which is uniformly continuous on X is also continuous on X

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Q: (b) If ƒ is continuous on (a, b) then there is a number c in (a, b) such that f(c) is an absolute…

A: Consider the statement  If f is continuous on (a,b) then there is a number c in  (a,b) such that…

Q: (5) Prove that Lp(f)<Uq(f) for any bounded function f and partitions P,Q of [a, b]. (Hint: Consider…

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

Q: กำหนดให้ f(x, y) = xy cos.x x² +3y²

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Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: A function f: R --> R is said to be continuous at c in R if for all epsilon > 0 there exists a delta…

A: Solution :Continous of a function at point 'c'A function fIR→IR is said to be continous at a point c…

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: Consider the given function:

Q: ery differentiable function y=f(x) defined

A: Given statement is: Every differentiable function y=f(x) defined on an open interval (a,b) has an…

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: g(x) = 0 for 7 distinct values of x in (a, b).Drawn a rough sketch of the case

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b). Suppose that…

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Q: A bounded function f which has only a finite number of points of discontinuity in [a,b] is…

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Q: Use the definition of uniform continuity (i.e., the e, d definition) to prove that f(x) = x² – x is…

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Q: Prove Fermat’s Theorem for the case in which f has a localminimum at c .

A: It is given that f has a local minimum value at c. Thus, by the concept of local minimum, fc≤fx when…

Q: f(x)=xmodx for all x  in R. Discuss the derivability of f(x) at x=0

A: Given:fx=xxwe have to find derivability of fx at x=0

Q: Suppese that X, X and that f is continuous. Show that f(X,) 1(X). Show w that ETLJY

A: Since you have asked multiple questions, we will solve the first Question for you. If you want any…

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

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Q: Q\ Use appropriate functions to guarantee that (). x sin if x # 0 h(x) = 0, if x = 0 is continuous…

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Q: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set. Is this statement…

A: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set  Given statement is…

Q: a) Show that, any function f: X → Y is continuous if X have discrete topology.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (1) Give an example of a function f such that | f |is Riemann integrable but f is not.

A: Consider the function f on the interval 0,1 defined by:  fx=1, if x is rational-1, if x is…

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: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

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Q: ® let f:X→R Cunt f X is bounded $ clused then f is uniformly conto

A: We have to prove that f is uniformly continuous if f is continuous, bounded and closed.

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

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Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

Q: Consider the sequence of functions {f} defined on the interval [0, 1], where fn(x) = nx Evaluate lim…

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Q: Prove that the extreme value theorem doesn’t work over ℚ. That is, find a continuous function, f: ℚ…

A: Consider the function f: ℚ→ℚ, where f(x)=sin(x). Consider the function in the closed interval 0, 2…

Q: Suppose that f(x) is convex on an interval I. Show that f(x1) + f(x2) + ·+ f(xn) n for arbitrary n…

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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: Let f be in R[a, b]. Prove that -- f is in R[a, b] and ∫a b -- f(x)dx = -- ∫ab f(x)dx.

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Q: II. Find all values of a and b that make -а - х, f(x) = {bx² + 1, Vx + b, x<-2 -2 sx<1 x21…

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Q: (5) Suppose that fis continuous on [a, b]- f(x) ≥ 0 Vx = [a,b] and [f(x)dx = 0. Prove that f(x) = 0…

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Q: 18.Q. Is it true that function on R? to R is continuous at a point if and only if it is both upper…

A: exercise 18.Q