Let a, b ∈ R with a < b and let f : [a, b] → [a, b] be continuous. Show that there exists x0 ∈ [a, b] such that f(x0) = x0.

Q: ) Prove that if f: RR is continuous at a, then so is f.

A: Continuous function:                                      A function f from R to R is…

Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Let f be continuous from [0, 1] to [0, 1]. By posing 6(x) = f(x) – x, show that fhas a fixed point…

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Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

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Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

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Q: Let I be an interval and let f : I → R be a strictly increasing function such that f(I) is an…

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Q: Let f : [a,b] → [a,b] be continuous. Prove that there exists x ∈ [a,b] such that f(x) = x.

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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: Show that the function f: C → C given by f(z) = z2 is continuous on C.

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Q: Prove that the function f: R → R: x x² + ax + b where a, b E R is continuous at every c E R.

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Q: Let ƒ be continuous and nonnegative on [a, b]. Show that there is c € [a, b] with f(c) = (±a få…

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Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

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Q: Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f : (a,…

A: Given:f:a, b→ℝ is an unbounded function. To show : f' is also unbounded On the contrary,  assume…

Q: Let I  be an interval and let f : I → R be a continuous, injective function. Show that f is…

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Q: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

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: f : R → R+ is a function such that f(x y) = f(x) − f(y) for all x,y ∈ R+. Prove thefollowing (a) f…

A: Let the given function is defined as f (x + y) = f (x) – f (y). find the continuity of the function…

Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

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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: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

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Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

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Q: Let f be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

A: Let f be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

Q: Let f: A → B be a function and let a ∈ A and b ∈ B. Does this mean it is true that f(a) = b? If not,…

A: We use definition of function .

Q: Let I be an interval and f : I → R be a function with Darboux property such that f(I) is finite.…

A: Let I be an interval and f : I → R be a function with Darboux propertysuch that f(I) is finite.…

Q: f(c) = g(c).

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Q: Consider the function ƒ : R → R defined by 2x + 5, ax² + x, Show that there is a value of a for…

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Q: Prove that f(x) = x - x is continuous at all points c in R.

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Q: Give an example of a continuous function f : C -> C, where f is holomorphic at 0 but no other point…

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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.