Real Analysis/Limits and Continuity Exercises

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Exercises

These problems are on the difficult or, to put it differently if not mildly, non-standard type. Try to work the problems without the hints because most times, you might have a different approach or way of thinking about a problem. Use the hints only if you are truly stuck! Without further ado, here are the problems:

1. Prove that the function, f(x) = 1/x is not uniformly continuous on the interval (0,∞).
2. Prove that a convex function is continuous (Recall that a function $f: (a,b) \rightarrow \mathbb{R}$ is a convex function if for all $x,y \in (a,b)$ and all $s,t \in [0,1]$ with $s+t = 1$, $f(sx+ty) \leq sf(x)+tf(y)$)
3. Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is $\exists p \in [0,1]$ such that $f(p) = p$
4. Prove that the space of continuous functions on an interval has the cardinality of $\mathbb{R}$
5. Let $f:[a,b] \rightarrow \mathbb{R}$ be a monotone function, i.e. $\forall x,y \in [a,b]; x \leq y \Rightarrow f(x) \leq f(y)$. Prove that $f$ has countably many points of discontinuity.
6. Let $f:(a,b) \rightarrow \mathbb{R}$ be a differentiable function, and suppose there is some positive constant $K$ such that $|f'(x)|\le K$ for all $x \in (a,b)$. (a) Prove that $f$ is Lipschitz continuous on $(a,b)$ (Hint: Use the mean value theorem). (b) Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function $f$ you are working with is uniformly continuous).

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