54. Let f:R" R and g:R Rbe continuous functions. Define h: R" → R by h(x) = glf(x)]. Show that h is continuous. Is it possible for h to be continuous even if f and g are not?

Q: 4. h(x) = x-T at X %3D 2. 5h(X) = x + x4-1 at X = %3D

A: Continuous function at x=a

Q: Define f : R → R by f(x) = 0 if x∈ R\Q f(x)=1/q if x=p/q with (p, q) = 1. Prove that f is…

A: Define f:ℝ→ℝ byf(x)=0if x∈ℝ\ℚ1qif x=pq with (p,q)=1Prove that f is continuous on ℝ\ℚ and…

Q: Is the function continuous on the interval (0, T COS x O no O yes O cannot be determined

A: Given problem:- Is the function ex/cosx continuous on the interval [0, π] ?

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

A: Given problem:- If f is not piecewise continuous on [0, ∞),then ℒ{f(t)} will not exist.  True…

Q: Let f(x) = { *, if x= 2 ; for the eax if x + 2 function to be continuous at x = 2 the value of a is

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Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

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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: 3. Use the definition to show that the function f(x) = is continuous on the interval [c, o0), where…

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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: If f(x) is continuous at x = c then f'(c) exists. True False

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Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

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Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

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Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

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Q: Suppose f is continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

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

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Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

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Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

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Q: Let f and g be continuous funCHons such that So f(x) dx =9,S f(x) dx = 5, and S g(x)dx= -7 what is…

A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

Q: B) If f is differentiable at x, then prove that f is continuous at x Is the converse true? Why

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Q: Suppose X is continuous with f(x) = x/2 from 0 to 2. What are the values of f(1) and P(X=1)? 1/2 and…

A: Given,f(x)=x2 ; 0<x<2

Q: Explain why or why not f(x) is continuous or not continuous at x = 1 using the ideas of the delta…

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Q: Assume f(x) is continuous on [a, b]. Prove that if f(x) > 0 for all x € [a, b], then S. f(x) dx > 0.…

A: Extreme Value theorem states that if a real-valued function f is continuous on the closed inteval…

Q: Let f(x) = CX– 2, x > I lx+ 8, x < 1 For what value of c is f(x) continuous on (-00, 0)

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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,…

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Q: Suppose that f(1)=8, f(4)=−4, f′(1)=−3, f′(4)=2, and f′′ is continuous. Find the value of ∫from 1 to…

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Q: Show that the following function is not continuous at (0,0). { 3 x2+y6, 0, , if (x, y) # (0,0); if…

A: To prove that given function is continuous. gx,y=xy3x2+y6,   if x,y≠0,00,              if x,y=0,0

Q: (d) f(x) = arctan x on R. (e) f(x) = 1/(1+ x²) on R.

A:

Q: 14. (a) If f and g are continuous functions on [a, b] with f> f g for every xe [a, b], must it be…

A: Solved below

Q: Suppose f is a continuous function defined on (-e, e) for some e > 0 and assume that f(0) = 0 and…

A: Since function f(x) is continuous at x=0. Also given that limx→0f(x)x=C

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

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Q: If f is a continuous function on [a, b], ƒ is differentiable on (a, b), and if both f(a) and f(b)…

A: This question is based on Rolle's theorem which is explained in step 2.

Q: 2. Prove that if f: I R is continuous on I, then |f|: I – R where |F(x) := |f(x)| Vz E I is…

A: f is defined on the set of real numbers. The detailed solution is as follows below:

Q: 50 if x E Q 1 if x € R\ Q Let f(x)= Prove that y=f(x) is not continuous at any points a E IR.

A: Take p ∈ Q and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1 but f (xn)→ 0…

Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: 10. Let L : R² → R be given by 2²–y? if x + y # 0 L(x) = L(x, y) = I+y 2x otherwise Decide whether L…

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Q: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

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Q: alue of c that makes the following function continuous: S 1+ cx if x 2.

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Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

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Q: ) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and…

A: Continuity and differentiability

Q: Suppose f ≤ 0 , f is continuous on [a, b] and ∫ab f dx= 0. Prove f(x) = 0, ∀ x ∈ [a, b].

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Q: Show that x³, if x 0 is not continuous at x = 0. %3D

A: According to the problem, we have

Q: Show that f : (0,∞) → R defined by f(x) := 1/x is not Lipschitz continuous.

A: Definition:  Let S⊂ℝ and a  function f:S→ℝ is Lipschitz Continuous at x∈S if there is a constant C…

Q: Prove f (x) 1+ mx is continuous at x = 1, where m e R

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Q: Find a value of k making h(x) continuous on (0, 5). 0 < x < 2 2 < x < 5 h(x) = x +1,

A: A function is continuous if limx→a+ h(x)  = limx→a- h(x) = h(a) Now in the given question, function…

Q: Suppose f is continuous and F'(x) = f(x). J s(x)dx =3, f(x)dx = 4, [f(x)dx= 5, and F(1) =6, evaluate…

A: Here this is simple application of integration and differentiation please watch carefully.

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

Q: If f:R → R is continuous at both 2 and 4, then f must be continuous at some c E (2,4). Select one:…

A: Consider the function f:R→R, that is continuous at both the points 2 and 4. Let there be a point c…

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.