a4 - f is continuous at Show that there exists a constant c...

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Department of Mathematics and Statistics McGill University MATH 242 Assignment 4 Due in class, Friday November 6 Full credit can be obtained from correct answers to questions totalling 70. 1. (10 points) Let c R and f : R -→ R be such that lim x c f ( x ) 2 = L . (i) If L = 0 , show that lim x c f ( x ) = 0 . (ii) If L = 0 , show by example that lim x c f ( x ) may not exist. 2. (10 points) If f : R -→ R is increasing and unbounded show that one or both of lim x →∞ f ( x ) = and lim x →-∞ f ( x ) = -∞ holds in the sense of proper divergence. 3. (10 points) Assuming only that | sin( t ) | 1 , but otherwise from first principles, show that lim x 0 x sin( x - 1 ) = 0 4. (10 points) Decide whether the following statement is true or false and then give either a proof or an explicit counterexample. For every subset A Q , there exsits a function f : R -→ R which is continuous at a point x R if and only if x / A . 5. (10 points) Show that every polynomial function of odd degree has at least one root. 6. (10 points) Suppose that f : R -→ R has the property f ( x + y ) = f ( x ) + f ( y ) for all x, y R . Suppose also that
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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 f-1 ( { a } ) has either or 2 elements. 9. (50 points) Let f : R-→ R have the property that lim x → a-f ( 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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