1. For each of the following, either give an example matching the description of the function or explain why such a function cannot exist. (a) f: (0,1] → (0, 1) onto and continuous (b) f: (0,1) - [0, 1] onto and continuous (c) f: (0, 1] (0, 1) onto and continuous

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Q: The graph of f(x)= - x2-1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 ).co (-∞, – 1)U…

A: Given, fx = xx2-1  We have to find the interval in which graph of  fx = xx2-1 is concave up.

Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

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Q: 8 Examine holomorphism for given function f(z) = |z|

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Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ¥x, yER. Show…

A: Introduction: Continuity is a property of a real-valued function. When the limiting value of a…

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Q: 4. Use limits to determine the values of a and b that guarantee the function below is continuous.…

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Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

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Q: 67. Is the function given by g(x) 1 continuous over %3D x + 5 the interval (-4, 4)? Why or why not?

A: g(x)=1x+5

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A: f is continuous on [1,5]. f has no local maximum or minimum. 2 and 4 are the critical numbers.

Q: Suppose that ƒ is continuous in [0,1] and that f(x) E [0, 1] for all x. Show that f(x) = x for some…

A: Given that f:0,1→0,1 is continuous on 0,1. To show fx=x for some x∈0,1

Q: 5. If g is continuous on an interval [a, b] and has an extremum at a number c € (a, b), then the…

A: I give two examples for this statement.

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: Prove that there is a function g(x) that is not everywehre continuous, yet it is even. Prove that…

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Q: [0,1] 6. If f is continuous on • prove that

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Q: Theorem 8.1. The following are equivalent: (1) X is connected. (2) There is no continuous function f…

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Q: Suppose h is a function such that h(1) = -2, h'(1) = 2, h"(1) = 3; h(1) =-2, h(2) = 6, h'(2)= 5,…

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Q: Consider the function f(x) = In(x + 5) over the interval [1, 9]. Does the extreme value theorem…

A: The given function is f(x)=lnx+5 over the interval 1,9  The extreme value theorem states that if…

Q: 1. Let f be defined on an open interval (a, b) and assume that f has a local maximum or a local…

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Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ♥x, yER. Show…

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Q: Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0&lt;=ƒ(x)&lt;= 1 for…

A: Let us assume a function g(x)=f(x)-x Since both f and x are continuous on the closed interval [0, 1]…

Q: Sketch the graph of a continuous function fon [0,4] satisfying the following properties. f'(x)= 0 at…

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Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

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Q: 20. Use the following information to sketch the graph of the continuous func- tion y = f(x) whose…

A: First observe that the function is continuous and its domain is all real numbers.Start by plotting…

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: Define f(2) in a way that extends f(s) = 2 to be continuous at s = 2. %3D -4

A: The function is defined as fs=s3-8s2-4.

Q: 1 Show that f (x) = is Libschitz function on %3D x +1

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Q: 7. Suppose f is a continuous function on [a, b] and Sf(x)g(x) dx =0 for every continuous function g.…

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Q: Let f be defined by the following. S2x-3 f(x) = 6 - x if x < 3 if x 2 3 (a) Is f continuous at x =…

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Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

A: Theorem: Let f be a differentiable function on the interval I. Then, If f'x&gt;0 for all x∈I, then…

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: 3-The function f defined by 2. Is f is Continuous at x=| ?

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Q: Suppose that f(1) = 5, ƒ(4) = -8, f'(1) = 4, f'(4) = -2, and f" is continuous. Find the value of |…

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Q: 13. Let g be defined by V7x +2 V6x + 4 | g(x) = if x 2 - and x 2 x – 2 k , if x = 2 For what value…

A: The given functiong(x)=7x+2-6x+4x-2,if x≥-27 and x≠2k,if x=2

Q: alue of c that makes the following function continuous: S 1+ cx if x 2.

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Q: Explain why the following function satisfies the hypotheses of Rolle's Theorem on the given…

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Q: Suppose f(x) is a continuous on the interval [a,b] and c is a number in (a,b). If f"(c) is…

A: If f(c) is a local maximum then f''(c) &lt; 0 and if f(c) is a local minimum then f''(c) &gt; 0

Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

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Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

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Q: Let f(z) be a real valued function defined for satisfying f(1) = 1 and f'(2) = then lim f(z) =)…

A: Given function is, f'x=1x2+fx2 The above equation f'x is always positive which means fx  is always…

Q: 5) Show that the function f(x)=-÷+1 below is continuous at x = -1. x-1 2- -3 -1- -2 1.

A: A function f(x) is continuous at x=c if  We have to show that is continuous at x=-1.

Q: (i) Let ƒ be the function defined on (-o, 0) by f(x) = r$(3 – x)}. (2 – x) -2 Given that f'(x) = and…

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Q: Suppose f(x) is a bounded real - valued function on [a.b) such that fela,b] . Does it follows that…

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Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: draw the graph of a function f on [0, 4] with the given property. Infinite one-sided limits at x = 2…

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