p1 - x E and y E c . They are connected by the segment [ x...

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #1 (due on W, September 15). Updated on Sat, September 11. 50 points are divided between 5 problems, 10 points each. #1. Let F be a compact subset of R n . Show that there are point x 0 , y 0 F , such that diam F := sup {| x - y | : x, y F } = | x 0 - y 0 | . Hint . There are sequences { x j } , { y j } in F , such that diam F = lim j →∞ | x j - y j | . #2. Let f ( x ) be a real function on a compact set E R 1 . Show that f is continuous on E if and only if its graph Γ = { ( x 1 , x 2 ) R 2 : x 1 E, x 2 = f ( x 1 ) } is a compact set in R 2 . Hint . You can use the fact that f ( x ) is continuous on E ⇐⇒ From { x 0 , x 1 , x 2 , . . . } ⊂ E and x 0 = lim x j it follows f ( x 0 ) = lim f ( x j ). #3. Let a subset E R n be nonempty, open and closed simultaneously. Show that E = R n . A possible way. Suppose that both E and E c are nonempty. Then there are points
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Unformatted text preview: x E and y E c . They are connected by the segment [ x , y ] := { x ( t ) := x + t ( y-x ) : 0 t 1 } in R n . Finally, consider the point z := x ( t ) , where t := sup { t [0 , 1] : x ( t ) E } . In the remaining two problems, the following notations are used: A c := R n \ A, o A-the interior of A, A-the closure of A. #4. For an arbitrary set E R n , show that ( o E ) c = E c . #5. Denote o E- the interior of E , E- the closure of E , o E- the interior of E , etc. Give examples of sets E such that: (a) o E 6 = o o E , (b) E 6 = o E....
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