1. If f:[a,b]->R is a bounded function such that f(x)=0 except for x E {c1, c2,..., cn} C [a,b]. Show that f is Riemann integrable on [a,b] and determine the Riemann integral value.
2. Suppose that f is any bounded function on [a,b] and that, for any number c E (a,b) the restriction of f to [c,b] is Riemann integrable. Show that f is integrable on [a,b] and that Integrate[a->b] f = Limit[c->a+] Integrate[c->b] f
3. Prove that the function f:[0,1]->R defined by f(0)=1, f(x)=0 if x is irrational, and f(m/n) = 1/n, if m,n E N are relatively prime is Riemann integrable.
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