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Unformatted text preview: f is continuous at . Show that there exists a constant c such that f ( x ) = cx for all x ∈ R . 7. (10 points) Let f : [0 , 1]→ [0 , 1] be continuous. Show that there exists x ∈ [0 , 1] such that f ( x ) = x . 8. (20 points) Show that there does not exist a continuous function f : [0 , 1]→ R with the property that for all a ∈ R the set f1 ( { a } ) has either or 2 elements. 9. (50 points) Let f : R→ R have the property that lim x → af ( x ) = f ( a ) . Show that there is a countable subset A ⊂ R such that f is continuous on R \ A . Please ﬁll your name and student number this cover sheet if you have answered questions worth more than 70 points. Attach the cover sheet to the front of your assignment. Family Name: Given Names: Student Number: Do not write in this box 1 /10 2 /10 3 /10 4 /10 5 /10 6 /10 7 /10 8 /20 9 /50 score...
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 Fall '09
 Drury
 Statistics, Continuous function, Department of Mathematics, Statistics McGill University

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