Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

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Q: Suppose that f is differentiable on some open interval containing c, and (c,f(c)) is a point of…

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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: Define g(-5) in a way that extends g(x)= 9(-5) = x² - 25 x + 8x + 15 to be continuous at x = -5.

A: We have to define g(-5) such that g(x) is continuous at x = -5.

Q: A function f is said to have a removable discontinuity at (a) if:1. f is either not defined or not…

A: We have to factor the numerator  At x=3 both of the numerator and denominator are zero, so we get…

Q: For all x in the domain of f, f(-x) = f(x) Functions with this property are even functions Verify…

A: Note: in the above image, we have sum total of two questions which are the following.  (i)     (ii)…

Q: Use the IVT to show that if f is continuous and one-to-one on an interval [a, b], then f is either…

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Q: If f is a rational function, then f is continuous for all real numbers. True O False

A: General form of rational function  p(x)q(x,    where q(x)≠0

Q: Sketch the graph of the function and use it to determine the values of a for which limit of f(x)…

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Q: Show that the function f(x) = x' + 3x² is continuous at every real number a.

A: fx=x3+3x2

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: The function is on the interval くx< (segment 1), on the interval くく |(segment 2), and

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Q: if x is rational, Consider the function f : R → R defined by f (x) = { otherwise. Determine the set…

A: We know that

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

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Q: Find a function that is defined for all real numbers and has f'(x) that exists for all real numbers…

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Q: Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval…

A: It is given that (c,f(c)) is a point of inflection of the function f.

Q: Determine whether the below function f is continuous or discontinuous at a =-3. If f is…

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Q: Find the domain of analyticity for g(z)

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Q: Determine whether the function of f(x) is continuous at x = -1 x2 -1 f(x) ={ x +1 X + -1 -2 X = -1

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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: Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove…

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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: Show that the function Arg(z) is not continuous on each point of the non-positive real axis.

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Q: Suppose that f: R→R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

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Q: Define g(-1) in a way that extends g(x) = 1 to be continuous at x= - 1. x² -x-2 2 9(-1)-D

A: Given that: g(x)=x2-1x2-x-2

Q: 2. Determine whether f is continuous or discontinuous at 1. (x – 1 if x > 1 f(x) -x² if x <1 O f is…

A: Given query is to find function is continuous or not at x = 1.

Q: B. Determine whether f(x) (x - 3) (2x + 1) (x + 4) is continuous at x= 3.

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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: Find x for which f(x) is not continuous: 3 x-1 f(x) =-+ x'x2 -1

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Q: Suppose that f: R R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

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Q: Find the absolute extrema of f on the given closed interval. 1. f(x)= x³ + 3x² - 9x ; [-4,4]

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Q: Prove if f and g are both functions that are continuous at some point a, then the difference f - g…

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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: Determine the intervals on which the following function is continuous. x2 - 3x +2 f(x) = x2 - 1

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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…

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Q: f(x)=- %3D x² +6x+9

A: A function f(x) is said to be continuous at a point 'a' when the function is defined at that point…

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: x 2

A: Consider the function f(x)=x-1              ,       x≤02x2-3x+1    ,       0&lt;x≤23x3-x2          ,…

Q: Show that f is a discontinuous function for all x, where f (x) is defined as follows: Show that f 2…

A: Here given function as fx=1for x rational-1for x irrational Also  f2x=fx·fx

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: Let f(x) = 2x² + 5x - 7 x-1 = 1 and determine what value for f(1) would make Show that f(x) has a…

A: Solution of given question is written in below steps.

Q: Suppose f and g are functions with domain R. If both f and g are even but ƒ +g is odd, then prove…

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

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.