Observe that f (x) = x 2 is continuous on [0, 1] with f(0) = 0 and f(l) = 1. Because f (0) < 0.5 < f(l), the Intermediate Value Theorem guarantees there is a c E [0, 1] such that f (c) = 0.5.

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

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

Q: Define g : R →R x sin (1/x), g(x) = 10, x # 0 x = 0. Explain why g is continuous at 0; in…

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Q: Find the x-values (if any) at which f (x)= 1s not continuous. x²+ 3x O fx) is not continuous at x =…

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Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

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…

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Q: VI. Suppose h' is continuous at x = 0 and h'(x) =tan(2x) - 1 if x > 0. Find h(0), if it exists.

A: Let's find.

Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

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Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: We have to find f(c)

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Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: Suppose f is continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

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Q: iii. is continuous at a provided g(a) # 0.

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Q: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) &gt;0. Does f have a local…

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Q: Suppose that f (x) > 0 for all x, f(x) has a continuous derivative and that е — 1 f(-3) = 0 and f(4)…

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Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: The answer is that f ''(c) tells us the concavity of f(x) at x = c. If f(x) is concave up, i.e., if…

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Q: Suppose that f: [0, ) – R is continuous, and that f(x) – l as x → - x. Prove that f is uniformly…

A: See the attachment

Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [0,…

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Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

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Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

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A: Since function f(x) is continuous at x=0. Also given that limx→0f(x)x=C

Q: 2. Show that the function f(x) = cos x is uniformly continuous on

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Q: Find a value of k making h(x) continuous on [0, 5]. kx , 0 < x < 1 h(x) = x +3, 1<x < 5

A: Given: hx=kx,0≤x&lt;1x+3,1≤x≤5 given function will be continuous over 0,5, if it will be continuous…

Q: Consider g(x) = rº/3. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: We find g'(x) using the power rule  Then we find the left and right-hand limits of g'(x) and g'(0)…

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Q: Suppose that f has continuous derivative and is twice differentiable in R. Suppose that f(a) = f(b)…

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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: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

Q: Let f(x) be a continuous function on [0, 1] and consider the limit 1 Xj +... f + In · drn. lim dx1…

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Q: (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies…

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Q: Find a constant b such that h is continuous at x = 2, where x +1 for |x| 2 With this choice of b,…

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Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

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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: 1. Use the e-ổ definition to prove that f(x) = x² is uniformly continuous on [a, 이.

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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.