Show that f is a discontinuous function for all x, where f(x) is defined as follows: for x rational -1 for x irrational f(x):

Q: Any rational function is continuous wherever it is defined; that is, it is continuous on its…

A: We know thatPoint No.1: the composition of continuous functions is continuous.Point No.2:…

Q: Let f (x) be a function satisfyingIf (x)| <x? for -1<x< 1. Show that f is differentiable at x = 0…

A: The solution is in step 2

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

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

A:

Q: Given that, f(x) = 2x3 - 3x2 - 12x + 1. F (a) the intervals on which f(x) is increasing

A: Given: f(x)=2x3-3x2-12x+1

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: Suppose that a function ƒ satisfies the following conditions for all real values of x and y: i) ƒ(x…

A: The given conditions are,

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

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Q: Show that the following function is continuous or discontinuous at. f(x)=x2+x-6/x2-4

A: The given function is

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: 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: Let f(x) 2x2+3 •given ɛ > 0,find M x2 such that if x > M we have |f (x) – 2| < ɛ. Conclude that…

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

A: It is given that f(x) is a continuous function and f(a)f(b)&lt;0. Then f(a) , f(b) have opposite…

Q: Suppose f(x) is a twice-differentiable function which is positive and decreasing for all real…

A: Consider the given conditions for f(x) and assume the graph for function f(x) for the given…

Q: Suppose that f(x) = 2 for x &lt; 3 and f(x) = -4 for x &gt; 3. (a) What is f (3) if f is…

A: For x&lt;3,f(x) = 2For x&gt;3,f(x) =-4Part (a) - f(3) = ? when f(x) is left continuous.f(3) =…

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: Consider the function f(x) = 8x + 9x-. For this function there are four important intervals: (-o,…

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Q: ,for x 1 Discuss the contuniuty of f andf'.

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Q: let be the rational function f be define as R(X) as R(x)=(/sqrt(x)-2)/(x-1) find the values of x…

A: We set R(x)=1 Then simplify

Q: For all x in the domain of the function +1, this func- tion is equivalent to: F.- G. 1 H. J. 1 x-x…

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Q: Let f: R R and let f(x) >0 for all xER. Show that f(x) is strictly increasing if and only if the…

A:

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: For f(x) = |x-1|, use the definition of differentiability f'(c) = lim determine if f(x) is…

A: Given fx=x-1

Q: Suppose that f:R → R is a continuous odd function, such that 1 f(x) dx = 5 and -1 | f(x) f(x) dx =…

A:

Q: Does there exist a function f such that f(0)  =-1, f(2)  =4, and f'(x) ≤ 2  for all  x?

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Q: Consider the following f(x)= 2/4-x2 Find the x-values at which f is not continuous. Which of the…

A:

Q: True or false: For the function f(x) =1/(x - 3), f is not defined at 3, so f is discontinuous at…

A: True or false: For the function f(x) =1x-3, f is not defined at 3, so f is discontinuous at 3.

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

A:

Q: Does there exist a real number c that makes the following functions continuous over (−∞,∞)? If…

A: If function is continuous at point of given interval, then

Q: For the function f(x) = Ax2 + Bx + C, determine the value of c guaranteed by the Mean Value Theorem…

A: f(x) = Ax2 + Bx + C f'(x)=2Ax+B By Mean value theorem:

Q: For f(x) = |x - 1|, use the definition of differentiability f'(c) = lim Ax+0 f(c+Ax)-f(c) Ax to…

A: Given: fx=x-1

Q: If f(x) =x²-x over x-1 and g(x) =x is it true that f=g?

A:

Q: a. Show that f has a removable discontinuity at x = 3. f(x) = x²7x+12 x + 2 x = 3

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

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Q: Is it possible to determine f(7) if f(x) = 3 for all x &lt; 7 and f is rightcontinuous at x = 7?…

A: Solution: Consider the given that f(x) = 3 for all x &lt; 7 f is right continuous at x = 7  The…

Q: If f is C2 on an interval prove that f(2 + k) – 2/(2) + {(z =A) = r"(2). - lim h-0 h2 = f"(x). The…

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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: 3. For f(x) = x' + 14, find all x such that f(x) = -13.

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Q: The piecewise function f(x) is defined below. f(x) = {kx² - 4, Is-2 x+ 2, xS-2 x>-2 For what value…

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Q: sketch the following rational functions by determining all critical values. f(x)=3x2-x3/x2-9

A: The given function is f(x) = 3x2-x3x2-9,This function is not defined at x = -3, x = 3.We try and…

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: Find the intervals on which f is increasing and the intervals on which it is decreasing. ​f(x)=6−x2

A: To find the function increasing or decreasing If f′(x)&gt;0 on an open interval, then f is…

Q: Let f be the function Зх + 5 for x 5 х Compute the left- and right-hand limits at x = 5 and…

A: Now, Left hand limit at x = 5 is, And right hand limit is,

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: Consider the function f(x) = 6x² – 2x + 11, 0 < x < 6. The absolute maximum of f(x) (on the given…

A: This question can be solved using the concept of absolute maxima and absolute minima.

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.