Uniformly Continuous
- صفحه اصلی
- Uniformly Continuous
Answer to Show that if f and g are uniformly continuous on A Reflect on what it means for a function to be uniformly continuous: For any epsilon greater than 0, there exists a delta greater than 0 such that if the absolute difference between any two points x and y in set X is less than delta, then the absolute difference between the function values at those two points (f(x) and …
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Then f + g is (b) → R and g: R → R are each uniformly continuous. Then fg is Suppose f:R uniformly continuous. → R are each uniformly continuous. Then fog is Suppose f:R → R and g: R uniformly continous. Suppose f:R + R, and suppose that lim h→0 (f (2 +h)-f(x for every x € R. Then f is continuous.
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Prove that if f is uniformly continuous on [k, ∞) for some k, then f is uniformly continuous on [0, ∞). B) Use (A) to prove that √x is uniformly continuous on [0, ∞). There are 2 steps to solve this one.
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Prove that if f is uniformly continuous on [k,∞) for some k ∈R, then f is uniformly continuous on [0,∞). Let f be a continuous function on [0,∞). There are 2 steps to solve this one.
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(c) Show that f/g is uniformly continuous on S if S is compact and g has no zeros in S. (d) Give examples showing that the conclusion of (b) and (c) may fail to hold if S is not compact. (e) State additional conditions on f and g which guarantee that fg is uniformly continuous on S even if S is not compact. Do the same for f/8.
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A uniformly continuous function of a uniformly continuous function is uniformly continuous. State this more precisely and prove it. Analysis problem, please type or write clearly, thanks!!!
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(a) Let f be a continuous function on [0, infinity). Prove that if f is uniformly continuous on [a, infinity) for some a > 0, then f is uniformly continuous on [0, infinity). (b) Prove that f(x) = Squareroot x is uniformly continuous on[0, infinity). Note that f' is unbounded on (0, infinity) (cf. Theorem 19.6).
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Question: (a) Show that continuous functions on compact sets are uniformly continuous. (b) Show that the uniform limit of a sequence of uniformly continuous R valed functions is uniformly continuous. (c) Consider the sequence of functions fn:[0,1]→R defined by fn(x)=⎩⎨⎧n2x22−n2x200≤x≤n1n1≤x≤n2n2≤x≤1 (i) Show that for each ...
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2. (a) (*) Show that a uniformly continuous function preserves Cauchy sequences: if f: A → R is uniformly continuous on A and (x n ) ⊆ A is a Cauchy sequence, then (f (x n )) is a Cauchy sequence. The point of part (a) is that the set A can be completely arbitrary. In particular, it …
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Give an example of a set that is not compact, but eve function continuous on that set is uniformly continuous. 25. Give an example of sets A and B and a continuous function f: A ∪ B → R such that f uniformly continuous on A and uniformly continuous on B, but not uniformly continuous A ∪ B. 26. Let E ⊂ R.
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