If f and g are continuous at c, then fg is continuous at c. Select one: O True O False

Q: If f is continuous on A) True B False on [a,b], then Sº [ f(x a xf(x) dx = x f(x) dx

A: It will solve by part method integration  ∫u v dx = u∫v dx −∫u' (∫v dx) dx u is the function u(x) v…

Q: Suppose f : [a, b] → [a, b] is continuous. Prove there exists c E [a, b] such that f(c) = c.

A: Now given f : a,b→a,b continuous gx=fx-x⇒ gx is continuous By Algebra of continuousga=fa-a ≥0gb=fb-b…

Q: Assume h : R → R is continuous on R and let K = {x : h(x) = 0}. Show that K is a closed set.

A: Given Data The closed set is K={x:h(x)=0}.   Consider a limit point y of (x:f(x)=0). There is…

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

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Q: If f and gare continuous on [a,b]and g(x)>0 for x E[a,b]- 1. Show that there exists cE[a,b] such…

A: (1) First mean value theorem:- If f:a,b→R and g:a,b→R be both interable on a,b and gx has the same…

Q: Answer true or false. If f is not piecewise continuous on [0, co>), then {f(t)} will not exist. True…

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Q: Answer true or false. If f is not plecewise continuous on [0, 00>), then {f(t)} will not exist. O…

A: Note that, L{f(t)}=F(s)=∫0∞e-stf(t) dt...(1)

Q: If f is continuous on [a, b], then there is cE[a, b] such that f(c) = av(f). Select one: True False

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Q: If f & R[a, b], then f is not continuous on [a,b].

A: Given statement statement:-  If  f∉Ra, b, then  f is not continuous on a, b.

Q: Determine whether the statement is true or false. f(t) : e-t- et 2 is continuous everywhere. O True…

A: Given that f(t)=2e-t-et Now, e-t-et =0 gives1et=etor, 1=e2tor, et=1or, t=0

Q: Answer true or false. 117.R.003. If f is not piecewise continuous on [0, co>), then (f(t)) will not…

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Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: Suppose f" is continuous everywhere. For any constants a and b, show that xf" (x) dx = bf" (b) –…

A: Explained below

Q: Q2: If f(x) and g(x) are continuous on [a,b]and if f = [g Prove that 3c e[a,b] such that f(c) g(c).

A: From the given information. The functions f(x) and g(x) are continuous functions on [a,b]. The…

Q: For what value of b is             continuous at every x?

A: The given function is fx=-2,x≤-1ax-b,-1<x<13x≥1. Assume that the above function is continuous.…

Q: If f and gare continuous on [a,b] and g(x)> 0 for xE[a,b]· 1. Show that there exists cE[a,b] such…

A: We are given that f and g are continuous functions on closed interval [a,b]and g(x)>0 for x∈[a,b]…

Q: Answer true or false. If f is not piecewise continuous on [0, ∞o>), then L{f(t)} will not exist. O…

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Q: If f is continuous everywhere, then af(bx) dæ < af(bx) dx, where a and b are real numbers. -3 O True…

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Q: Iff is continuous at a point a and c e R, then cf is continuous at a.

A: Note that, f is said to be continuous at 'a' if limx→af(x)=f(a) And, (cf)(x)=cf(x)

Q: Suppose f and g are continuous on [a, b] and f f(x) dx = S% 8(x) dr. Prove that there is cE [a, b]…

A: Suppose f and g are continuous functions on [a, b] such that ∫abf(x) dx=∫abg(x) dx We need to prove…

Q: if f(x) is continuous on[a,b] then -S f{x)dx) = f(x) dx Select one: O True O False

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Q: Define f : R" - ||x || 1. Prove that f is continuous on R". (To prove that f is continuous at a…

A: For continuity left hand limit (LHL) = right hand limit (RHL) =value of the function at that point…

Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: O No, because f is not continuous on the closed interval [a, b].

A: Given that: The function f ( x) = x2 - 81x - 4     ,  [ -9, 9  ]  .

Q: Iff is continuous on [a,b], then [xf(x) dx = x = xf²f(x) A True B) False f(x) dx

A: We will solve the problem

Q: If f is continuous on [a, b], then there is cE[a, b] such that f(c) = av(f). Select one: O True…

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Q: If f(x) and g(x)are continuous on [a,b]and if [f = [8 . Prove that 3ce[a,b] such that f(c)= g(c).

A: Here, we will going to use Intermediate Value  Theorem to prove this result

Q: If f is continuous on [a, b], then ſ tf(x)dx =t f f(x)dæ Select one: O True O False

A: Given: f is continuous on a, b, then ∫t f(x)dx=t∫f(x)dx

Q: 8. PROVE: Suppose that SCR is connected, and that f : R → R is continuous. Then f(S) is connected.…

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Q: If f is continuous on [a, b] and f(x) 20 for all x E[a, b], then Select one: O True O False

A: Let f(x)=x, x∈2, 3 I=∫23f(x) dx =∫23xdx =x323223 =23x3223 =23332-232 =2333-22

Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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Q: If X is continuous on the interval 0 to 10, t O True False

A: For a continuous distribution, and there exist infinite points.

Q: If n is a positive even integer and f(x) = Vx, then f is continuous everywhere. True False

A: Given that n is a positive integer and f(x)=(x)^1/n.

Q: B/ For what value of a is fo continuous at every x? x? – 1, x< 3 { fx) = 2ах, x2 3

A: Our objective is to find the value of a for which f(x) is continuous at every x.

Q: If f is continuous and even on [-a, a], then f(x S'f)dx = [ °ft-x)dx Select one: O True False

A: f is continuous and even on [-a,a] , then ∫-aaf(x)dx=∫-aaf(-x)dx

Q: Given that f and g are continuous on [a, b], that f(a) < g(a), and g(b) < f(b), show that there…

A: Explanation of the answer is as follows

Q: If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

A: Proof:            Given: f' is a continuous…

Q: Determine whether the statement is true or false. If f(x) is continuous at x = c, then so is f(x).…

A: Solution is given as below:

Q: The Fundamental Theorem of Calculus says that f(x)da = F(a) – F(b) provided f = F' Is continuous on…

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Q: Suppose the function ƒ has the property that IS(x) – S()| < |x -|| for each pair of points x, t in…

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Q: If f is continuous on [a, b] and f'(x) 0 O f(b) f(b)

A:

Q: If f(x) = is continuous at x=, Find A (Ar? Ax x2

A: Given: fx=x3,x&lt;12Ax2,x≥12 We know that any given functiony=fx is continuous at x=a If,…

Q: 4. Let f:A-→R and g:B–→R where f(A) CB. Explain to prove that if f is continuous at Ce A and g is…

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Q: 2. Do as indicated. x +1 if x m,

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Q: Assume that f is continuous in [a, b], twice derivable in (a, b), and that f (a) = f (d) = f (b) = 0…

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Q: If f(x) is continuous everywhere, then square root of f(x) is continuous everywhere. * O TRUE O…

A: We have to find out if f(x) is continous everywhere then square root of f(x) is continuous…

Q: If f(x) is continuous with f(a)f(b) < 0, then the e Select one: O an indeterminable number of roots…

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Q: True o false ? If fand g g are Continuous on faib I, then

A: Please see the answer below.