Problems in Mathematics/To be added

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2 Exercise Suppose is infinitely differentiable. Suppose, furthermore, that for every , there is such that . Then is a polynomial. (Hint: Baire's category theorem.)

Exercise and are irrational numbers. Moreover, is neither an algebraic number nor p-adic number, yet is a p-adic number for all p except for 2.

Exercise There exists a nonempty perfect subset of that contains no rational numbers. (Hint: Use the proof that e is irrational.)

Exercise Construct a sequence of positive numbers such that converges, yet does not exist.

Exercise Let be a sequence of positive numbers. If , then converges.

Exercise Prove that a convex function is continuous (Recall that a function is a convex function if for all and all with , )

Exercise Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is such that
Proof: Let . Then

Exercise Prove that the space of continuous functions on an interval has the cardinality of

Exercise Let be a monotone function, i.e. . Prove that has countably many points of discontinuity.

Exercise Suppose is defined on the set of positive real numbers and has the property: . Then is unique and is a logarithm.