Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0 as x → 0, but the limit of f(x) as x → 0 does not exist. Show why

Q: Suppose that g: [a, b] R is a continuous function. Show that there exists a point c € (a, b) so that…

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Q: 5. Define f: R→→ R and g: R → R defined by f(x) = max{-x, x} and g(x) = |x|, respectively. Prove…

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Q: Let F be a continuous function on [a, c} , where a < b< e, and F(z) dz = 6, F(2) dz = 9 A. Evaluate…

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Q: Suppose f: [-1,0] → R and g: [0, 1] → R are continuous and f(0) - g(0). Define h: [-1,1] → R by…

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Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: Define a function f : Q → N of your choice which is surjective. Then prove that the function you…

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Q: Prove that a function f is continuous at a limit point p iff the limit as x goes to p is f(p).

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Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

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Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: For real numbers m and b with m + 0, show that the map f: R-→ R defined by f(x) = mx + b is…

A: Given m and b be the real number with m≠0 and f:R→R defined by fx=mx+b. We have to show that fx is…

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

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Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

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Q: 1). Does the left-side limit of the difference quotient of the function g(w)=|w| exist at w=0? If it…

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Q: Define f: R -R by Sir +1 if r 4. Find all real values at which f(x) is continuous. Prove that your…

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Q: Let X, Y and Z be sets. A, B are subsets of Y. f:X → Y is a function. Prove the given statement if…

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Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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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.