Midterm exam:
26 April, 2010
The Problem
Let f:[0,1]-->[0,1] be the function defined to be 0 at the irrationals and 1/q when x = p/q in reduced form.
1. For what points in x in [0,1] is f continuous? Prove your answer to this question, and all the following also..
2. For what points in x in [0,1] is f differentiable?
3. Does the integral of f over [0,1] exist?
Let F:[0,1]x[0,1] --> [0,1] be defined by F(x,y) = f(x)f(y).
4. For what points (x,y) in [0,1]x[0,1] is F continuous?
5. Is F differentiable at (0,0)?
6. Does the integral of F over [0,1]x[0,1] exist?
Due this semester.
Page 5: 2, 4, 6, 18.
Page 12: 11, 12.
Due Monday, 25 January.
Page 18: 2, 9.
Page 25: 5, 7, 8, 10.
Due Monday 1 February.
Page 32: 2, 10, 11.
Page 37: 2, 3, 4, 6, 7.
Due Monday 8 February.
Page 45: 1. 4. 6. 9. 17.
Due Monday 22 February.
Page 76: 16.
Due Monday, 1 March.
A. If X and Y are CW complexes, show that XxY is a CW complex.
B. Calculate the homology groups of S^1 x S^2.
C. Calculate the homology groups of the 3-torus.
D. Calculate the homology groups of real projective n-space.
Due Monday, 8 March.
Page 160: 5, 10, 11.
Page 172: 5, 7.
Due Monday 12 April.
Page 172: 8, 9, 10, 12.
Page 185: 2, 8.
Due Monday 19 April.