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A function which is not Riemann integrable

If for all irrational , for all rational x, prove that on for any . (PMA Ch.6. ex. 4)

Take any interval such that , then we can funish this interval with a partition such that and and for all . 32 more words

PMA.Rudin

A constant integrand

Suppose , is continuous on , and . Prove that for all . (PMA Ch.6. ex.2)

As is continuous on the fundamental theorem of calculus (6.21) applies, and for all . 29 more words

PMA.Rudin

A single discontinuity makes no difference

Suppose increases on , , continuous at , , and if . Prove that and that . (PMA Ch.6. ex.1)

By theorem 6.10, , however, we can also show this by taking the partition , where we assume WLOG , and such that . 32 more words

PMA.Rudin

Uniform differentiability

Suppose is continuous on and . Prove that there exists such that whenever , , . (This could be expressed by saying that is uniformly differentiable on if is continuous on .) Does this hold for vector-valued functions too? 82 more words

PMA.Rudin

An easy L'Hospital's

Suppose exist, , and . Prove that . (PMA 5.7)

and exist, hence the limits and as tends towards exist. Now as , and . Hence .

PMA.Rudin

Mean value theorem, continuity and monotonicity

Suppose (a) is continuous for (b) exists for (c) (d) is monotonically increasing. Put and prove that is monotonically increasing. (PMA 5.6)

To prove that is monotonically increasing we need to show that for all values of (this is a simple consequence of the mean value theorem). 73 more words

PMA.Rudin

Mountains into Molehills

We all have things in life that we don’t particularly love to do … sad to say that’s a thing that will probably follow us all the way through our life, whatever stage, how ever big of small that thing that we don’t want to do is. 483 more words