6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {|x – y| : y € A}. Show that d(x, A) = 0 if and only if x is an element of the closure A.

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6. The distance between a point x E R and a nonempty set A CR is defined by
d(x, A) = inf {\x – y| : y E A}.
Show that d(x, A) = 0 if and only if x is an element of the closure A.
7. Which of the following functions f: (0,1) → R are uniformly continuous? Explain.
f (x) = x³ + 3x,
f (x) = Vx,
f(x) = tan(1/x).
Transcribed Image Text:6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {\x – y| : y E A}. Show that d(x, A) = 0 if and only if x is an element of the closure A. 7. Which of the following functions f: (0,1) → R are uniformly continuous? Explain. f (x) = x³ + 3x, f (x) = Vx, f(x) = tan(1/x).
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