if x is rational, Consider the function f : R → R defined by f (x) = { otherwise. Determine the set of points at which f is continuous. You can assume without proof that every nonempty open interval contains a rational number and an irrational number.

Q: Suppose that f is continuous on R and that f(x) = 0 whenever a is rational. Prove that f(x) = 0 for…

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Q: Let f be a function defined on JO, 1[ by : 0, if x is irrațional f (x) ={1/q, if x=p/q; where p and…

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Q: Suppose f is a function which satisfies the properties: f(x + y) = f(x) + f(y) + x²y + xy? for all…

A: Given that, f(x+y)=f(x)+f(y)+x2y+xy2 for all real numbers 'x' and 'y'  and, limx→0f(x)x=1   As we…

Q: Find the largest open interval, centered at the origin on thex-axis, such that for each x in the…

A: f(x) = x +2 f(0) = 2 ∈ = 0.2   Now, | f(x)  -  f(0) | <  ∈

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: If for a real continuous function f(x), f(a)*f(b)<0, then in the range of [a,b] for f(x)=0, there is…

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Q: Let f be a function defined on JO, 1[ by : | 0, if x is irrational f(x) ={1/q, if x=p/q; where p and…

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Q: let f(x) =x2 , on interval 0<x<2n, sketch

A: We have to sketch the graph of the function f(x)=x2 on the interval 0≤x≤2π.

Q: Find the value of m such that g(x) = {x+5 , if x ≤ m} and g(x) = {2x - 3 , if x > m} is continuous…

A: Function limit and continuity

Q: If f is analytic in a neighborhood of ro and f(xo) = 0, show that f(r)/(x – ro) is analytic in the…

A: Analyticity

Q: Prove that the following function is continuous at 0. x, if x is rational f(x) = {0, if 0, if x is…

A: Given function is f(x) = x,if x is rational0,if x is irrational

Q: Consider the function x - 3 for x4 By using the characterization of continuous function in terms of…

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Q: If a function is real and continuous in the region from a to b and f(a) and f(b) have opposite signs…

A: Given a real function with f(a) and f(b) having opposite sign.

Q: Define the following real function: f(x) = 2 - x, x is rational and f(x) x2, x is irrational. What…

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Q: If F is continuous on the set of real numbers, then A True B) False

A: The solution is given in the next step.

Q: Let g : R → R be a bounded function and define f : R\{0} →R by 1 f(x) = g(x)+ Using only the…

A: It is given that ;                f : R \ 0 →R  defined by     fx = gx + 1x2 Where,   gx  is a…

Q: Define f:R → R by f (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given, f:R→R by f(x)=5x if f(x) is rational and f(x)=x2+6 if x is irrational.To prove that f is…

Q: Suppose f and g are real-valued functions defined on all of R, f is measurable, and g continuous. Is…

A: Given, that f and g are real valued function on ℝ and f is measurable and g is continuous function.…

Q: Let f be a function defined on ]0, 1[ by : 0, if x is irrațional f (x) = {1/q, if x=p/q; where p and…

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

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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…

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Q: Suppose that f is a function such that f' (c) = 1 for some number c in the interval [a, b]. Is it…

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Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. b) Prove that f is differentiable…

A: Given fx=x2x is rational0x is irrational To prove f is differentiable at exactly one point, namely…

Q: Define f:R → R by ƒ (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given :  f:ℝ→ℝ  defined by ;         f(x) = 5x,if x is rational.x2+6if x is irrational To prove :…

Q: Determine whether each of the following statements is true or false, and explain why. A rational…

A: Given that: A rational function is continuous everywhere. The Given statement is FALSE

Q: Let f be a function defined on ]0, 1[ by : 0, if x is irratțional f (x) ={1/q, if x=p/q; where p and…

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Q: Show that the function is continuous only at x = 0. (Assume that k is any nonzero real number.)

A: The given function is fx=0    if x is rationalkx   if x is irrational where k is nonzero. Note that…

Q: ind the values of c that make f continuous everywhere: { Sa² – c² if x 3

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Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let f be differentiable on an interval A. If f'(x) not equal to  0 on A,show that f is one-to-one on…

A: Given that, Let f be differentiable on an interval A. If f'(x) not equal to  0 on A. We have to show…

Q: Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one real…

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Q: Use the definition of continuity to determine whether f(x) = (x2 - 9)/(x + 3) is continuous at a=…

A: fx=x2-9x+3⇒fx=x-3x+3x+3 The common factor of numerator denominator is x+3. Therefore the given…

Q: 4. Suppose that f is continuous on a closed interval [a, b] and differentiable on the open interval…

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

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Q: Suppose that f is a nonconstant continuous function defined on an interval I. a) Prove that f(I)…

A: Given that f is a non-constant continuous function. Then there exists x and y in the interval I such…

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

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

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Q: Use the definition of continuity to determine whether f(x) = 3x2 - 2x + 1 is continuous at a=4.

A: Given f(x)=3x2-2x+1we have to use the definition of continuity to determine the given functionis…

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

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Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: Then the value of k for which f(x) is continuous for all real values of r is

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Q: ) Is every rational function on D holomorphic on D?

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Q: If f is a continuous function on a compact set, show that either f has a zero or f is bounded away…

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Q: Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?…

A: Given f(x) = ln(x+1)  , x≥03-x2     ,  x&lt;0

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

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Q: Use the definition of continuity to determine whether f(x) is continuous at a= 0.

A: Given function, We have to use the definition of continuity to determine whether f(x) is continuous…

Q: If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2

A: We can prove this with the help of Lagrange Mean value Theorem as follows.

Q: Show that f is a discontinuous function for all x, where f(x) is defined as follows: for x rational…

A: Let us consider a point x= a ,where a is rational, Now for the function to be continuous, at x=a,…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.