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'''2 Exercise''' ''Suppose <math>f</math> is infinitely differentiable. Suppose, furthermore, that for every <math>x</math>, there is <math>n</math> such that <math>f^{(n)}(x) = 0</math>. Then <math>f</math> is a polynomial.'' (Hint: Baire's category theorem.)

'''Exercise''' ''<math>e</math> and <math>\pi</math> are irrational numbers. Moreover, <math>e</math> is neither an algebraic number nor p-adic number, yet <math>e^p</math> is a p-adic number for all p except for 2.''

'''Exercise''' ''There exists a nonempty perfect subset of <math>\mathbf R</math> that contains no rational numbers. (Hint: Use the proof that e is irrational.)''

'''Exercise''' ''Construct a sequence <math>a_n</math> of positive numbers such that <math>\sum_{n \ge 1} a_n</math> converges, yet <math>\lim_{n \to \infty} {a_{n+1} \over a_n}</math> does not exist.''

'''Exercise''' ''Let <math>a_n</math> be a sequence of positive numbers. If <math>\lim_{n \to \infty} n \left({a_n \over a_{n+1}} - 1 \right) > 1</math>, then <math>\sum_{n=1}^\infty a_n</math> converges.''

'''Exercise''' ''Prove that a convex function is continuous <big>(</big>Recall that a function <math>f: (a,b) \rightarrow \mathbb{R}</math> is a ''convex function'' if for all <math>x,y \in (a,b)</math> and all <math>s,t \in [0,1]</math> with <math>s+t = 1</math>, <math>f(sx+ty) \leq sf(x)+tf(y)</math><big>)</big>''

'''Exercise''' ''Prove that every continuous function ''f'' which maps [0,1] into itself has at least one fixed point, that is <math>\exists p \in [0,1]</math> such that <math>f(p) = p</math>''<br />
Proof: Let <math>g(x) = x - f(x)</math>. Then 

'''Exercise''' ''Prove that the space of continuous functions on an interval has the cardinality of <math>\mathbb{R}</math>''

'''Exercise''' ''Let <math>f:[a,b] \rightarrow \mathbb{R}</math> be a monotone function, i.e. <math>\forall x,y \in [a,b]; x \leq y \Rightarrow f(x) \leq f(y)</math>. Prove that <math>f</math> has countably many points of discontinuity.''

'''Exercise''' ''Suppose <math>f</math> is defined on the set of positive real numbers and has the property: <math>f(xy) = f(x) + f(y)</math>. Then <math>f</math> is unique and is a logarithm.

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