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Answer ”True” or ”False” for each of the following: (i) If f,g : R ? R...

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Advance Math

Answer ”True” or ”False” for each of the following:

(i) If f,g : R ? R and are both continuous at a number c, thenthe composition function f ?g is continuous at c.

(ii) If functions h1, h2 : R ? R and are both uniformlycontinuous on a non-empty set of real numbers E, then the producth1h2 is uniformly continuous on E.

(iii) If a function g : R ? R, then there exists a function G :R ? R such that G0(x) = g(x) for all x ? R. (iv) If A ? B and A iscountable, then B is uncountable.

(v) If f : R ? R is continuous and positive at a number c, thenf is di?erentiable at c. (vi) If K is a non-empty compact set ofreal numbers, then either K is ?nite or uncountable.

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