A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| < C|x – a'|ª for some C, a > 0. Show that if 0 < a <1 then f is uniformly continuous in [a, b].

Q: Find continuous functions f1, f2, f3, f4 defined on the open unit interval I = (0, 1) (i.e.…

A: 1. f1(x) = (M1 - L1)x + L1, for all x in (0, 1)  This function is linear and maps (0, 1) onto (L1,…

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A:

Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x&lt;1           x…

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A: The given function is fx=x-5x2+6x-5. We have to find the point where the given function is…

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Q: Let f. ROR be a Continuous function with f(x) ≤ 10 for = all XER. Explain why f is not homeomorphism

A: A homeomorphism is a continuous bijection with continuous inverse.

Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

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A: This question is based on Rolle's theorem which is explained in step 2.

Q: If f is a continuous function on [a, b], and if f (a) = -2 and ƒ (b) = 2, then there exists a real…

A: Hi, you can find the answer in the handwritten form.  Given f(a)=-2  f(b)= 2

Q: Only one can be true

A: since f(a)&gt;0, so there is at least x=a , be the point at which function value is positive. Hence,…

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

A:

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.