3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing ro such that f(x) > 0 for all r E I.

Q: consider the function f defined on [0,∞),f(x)=(x^r)sin(1/x),for x≠0 and f(x)=0 and for x=0, where r…

A: Solve the following

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Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

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: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Suppose f:[0, 27] → R is continuous and f(0) = f(2n). Prove that there exists at least one point c E…

A: This question is based on intermediate value theorem .  The theorem says ,  If f is any continuous…

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

Q: Observe that f (x) = x 2 is continuous on [0, 1] with f(0) = 0 and f(l) = 1. Because f (0) < 0.5…

A: f(x)=x2 is continuous on [0,1] with f(0) = 0 and f(l) = 1 we need to prove that according to…

Q: Let f : R → R, and let x E R. Prove that f is continuous at x if and only if for every e > 0 there…

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Q: If f(x) is continuous in the interval [-2, 1], ƒ(−2) = −4, and f(1) = -1, can we use the…

A:

Q: For what value of b is X - b b + l' x 0 continuous at every x?

A: The given function is gx=x-bb+1if x<0x2+b if x>0. If the above function is continuous at x=0,…

Q: Suppose that the function f: [0, 1] in R is continuous, f(O) > 0, and f(l) = 0.Prove that there…

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

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Q: Suppose h' is continuous x>0. IF x = 0 and h'(x) = 3√tan (2x)-1 it exists. i at Find h (0), 'F

A: For the solution of the problem follow the next step.

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

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

A:

Q: Assume that the function f is such that f(3) = 1, ſ f(x) dx = 5, and f'(x) is continuous on [0, 3].…

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

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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: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

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 : R → R, and let x E R. Prove that f is continuous at x i and only if for every e > 0 there…

A: We have to provex that  epsilon delta definition of continuity:

Q: 7. If f is a continuous function such that f(t)dt = x sin x + f(t) dt for all x , find an explicit…

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Q: Set f(x) = xª for x ≥ 0 and f(x) = 0 for x < 0. For which values of a is f continuous at 0, and for…

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Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

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Q: Ler f : R → R be continuous. If f(c) > 0, prove there exists a neighborhood N of c such that f(N) >…

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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: 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: 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…

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

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

A: The solution is :

Q: suppose f (x) = Vx (x>0),prove that f is differentiable (0,00) and find f'(x) on

A: Function f(x) = √x (x&gt;0)  We know f(x) =Xn

Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.

A: The given statement is true. Because when f will be non negative then the derivative of the given…

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

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Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

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

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: 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: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

Q: 2. (i). Let g : R→R be continuous at c and let C ={r€R:f(x)=0}. If c, EC and lim c, =c. %3D Prove…

A: PART A Let g: ℝ→ℝ be continuous at c. Let C={ x∈ℝ : f(x)=0}  Let cn∈C and  lim cn=c  As,  cn∈ C…

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

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Q: 15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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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: Suppose that f(x) is a continuous function on the interval [0, T] . If f(x)=(sin x)ª for 0<x<T, then…

A: we have to find the value of given function. the solution is following.

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…