For each of the following statements, determine whether it is true or false. Give a counterexample IF IT IS FALSE (but you do not need to prove it if it is true). (a) Every continuous function f : (0,1) → R is uniformly continuous. (b) Every contimuous function f : [0, 1] → R is uniformly continuous. (c) Every uniformly continuous function f : D → R is continuous.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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For each of the following statements, determine whether it is true or false. Give a
counterexample IF IT IS FALSE (but you do not need to prove it if it is true).
(a)
Every continuous function f : (0, 1) → R is uniformly continuous.
(b) Every continuous function f: [0, 1] + R is uniformly continuous.
(c) Every uniformly continuous function f : D→ R is continuous.
(d) If the functions f : D→ R and g : D → R both are uniformly continuous, then so is f+g : D→R.
(e)
If the functions f : D →R and g : D → R both are uniformly continuous, then so is fg : D→ R.
(f) If the functions f : D→R is uniformly continuous, then so is af : D+R where a is any number.
(g) Every function f : N -R is uniformly continuous, where N denotes the set of natural numbers.
(h) Every function f : S +R is uniformly continuous, where S denotes a set of finitely many elements.
Transcribed Image Text:For each of the following statements, determine whether it is true or false. Give a counterexample IF IT IS FALSE (but you do not need to prove it if it is true). (a) Every continuous function f : (0, 1) → R is uniformly continuous. (b) Every continuous function f: [0, 1] + R is uniformly continuous. (c) Every uniformly continuous function f : D→ R is continuous. (d) If the functions f : D→ R and g : D → R both are uniformly continuous, then so is f+g : D→R. (e) If the functions f : D →R and g : D → R both are uniformly continuous, then so is fg : D→ R. (f) If the functions f : D→R is uniformly continuous, then so is af : D+R where a is any number. (g) Every function f : N -R is uniformly continuous, where N denotes the set of natural numbers. (h) Every function f : S +R is uniformly continuous, where S denotes a set of finitely many elements.
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