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# midterm - Midterm Math 215 Fall 2008 You may use Rudin for...

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Midterm — Math 215 Fall 2008 You may use Rudin for the exam. In your answers, you may cite results proven in Rudin. No other source may be consulted. You may not discuss the exam with anyone. Remember to sign the pledge. 150 minutes, 5 questions I. Consider the real numbers R with the standard Euclidean metric. For each of the three following functions f,g,h : R R , find the set of points in the domain R at which the function is contin- uous. (i) f ( x ) = x 2 + 1 6 x + 5 6 for x < 1 and f ( x ) = 3 7 x 7 + 11 7 for x 1. (ii) g ( x ) = 3 x 2 +1 . (iii) h ( x ) = x 2 2 x +3 if x is rational and h ( x ) = x +1 if x is irrational. For part (ii), recall x denotes the integer part of x (the greatest integer less than or equal to x ), so 6 . 9 = 6 , 7 = 7 , 7 . 1 = 7 . Answer with proof — however, you may use without proof that a polynomial function is continuous. II. Consider R with the standard Euclidean metric. Answer the follow- ing questions with proof.

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midterm - Midterm Math 215 Fall 2008 You may use Rudin for...

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