2. Prove: Let ƒ : R → R be a continuous function satisfying f(x) = 0 for all rational numbers x. Then f(x)=0 for all real umbers x.

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: Prove that f(x) = |x| is continuous at all points c in ℝ.

A: Continuity of a function can be done by many methods as,

Q: Examine if f(x) = x3sin(1/x) is uniform continuous on the interval (0,2]

A:

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

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: Let f(x)=x² -x² +x + 1 and g(x)= 3-х max{f(t):0sx; for 0 s x s 1} for 1 < x s 2 Discuss the…

A: Topic:- Functions

Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x&gt;c Now  limx→c+ f(x)=limx→c+…

Q: • Use the IVP to prove that r3 – 4z+2 has a zero in the interval [0, 1] • Use the IVP (and a similar…

A:

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

A:

Q: Let fbe a continuous function with domain R such that f(1) = 0.05. Prove that fis strictly positive…

A:

Q: Suppose that f is a function on the interval [1, 3] such that –1 < f(x) < 1 for all x and f(x) dx…

A: It is given that ∫13fxdx=0 and -1≤fx≤1.

Q: Suppose that f is continuous at 0, and that f satisfies f(x + y) = f(x)+ f(y) for all real numbers…

A:

Q: Let f : R → R be a continuous function such that f(q) = 0 for every rational number q. Prove that…

A:

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…

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: 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: 2. Consider the function if x = in lowest terms q > 0, p, q € Q; f(x) = O if x e Q°,x = 0 Question:…

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Q: Let f be a function defined on ]0, 1I[ by : 0, if x is irrational x= p/q; where p and q are positive…

A:

Q: Find all values of c such that f is continuous on (-00, 00). f(r) = { 8 – x², !! x>c

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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: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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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: Let f be a continuous function on R such that f(x+y)=f(x)+(y) ¥x, yER. Show that f(x)3cx for some…

A:

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: 8. Suppose that f is a continuous function with f(x) > 0 for all real numbers x, and lim f(x) = 0 =…

A: Consider the provided question, Suppose that f is a continuous function with f (x)&gt;0 for all real…

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

Q: Consider the function: { 0 ;if x is rational ; if x is irrational f(x) = Use a d, e argument to…

A:

Q: 1 - is continuous at a. f If f(x) 0 for all x in the interval I, and f is continuous at a in I then…

A: 1/f is rational function this is discontinous where either f is discontinous or f is zero  But f is…

Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. a) Prove that f is continuous at…

A:

Q: 3. Suppose f is continuous on an open interval containing x, and f has relative extremum at x = x,…

A: The solution is :

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

Q: Prove that if f is a continuous function with domain [0, 1] and X is the set of all x in [0, 1] such…

A: F is a continuous function with domain (0,1) and X is the set of all x in (0,1) such that F(x) = 0.…

Q: Find values of x, if any, at which f is not continuous. Prove using limits. f(x)= x/x^2 +4

A:

Q: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

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

A:

Q: 45. Let f : R → R be a continuous function, such that f(x)= 0 if x is a rational number. Show that…

A: Given continuous function is f: ℝ → ℝ such that f(x) = 0 if x is a rational number.

Q: Use the following outline to prove the theorem that every continuous function on a closed finite…

A: The given problem is related with real analysis. Here, we have to show that every continuous…

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

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Q: Give an example of a function f : R → R that is continuous at x if x is an irrational number, but it…

A:

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: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

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

A:

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

A: Given that f is a continuous function with domain 0,1. Let, X=x| x∈0,1 and fx=0. Also, given that s…

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

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

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.