Real Analysis/Limits and Continuity Exercises/Hints

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These are a list of problems for the Limits and Continuity section of the wikibook.

Unsorted 1

1. Although the wikibook asserts the truth of the following questions in this table, it is a good exercise to prove them. Thus, given the continuous functions $f$ and $g$, prove the following
   • $\lim _{x\rightarrow c}{(f+g)(x)}=f(c)+g(c)$
   • $\lim _{x\rightarrow c}{(f\cdot g)(x)}=f(c)\cdot g(c)$
   • $\lim _{x\rightarrow c}{\left({\dfrac {f}{h}}\right)(x)}={\dfrac {f(c)}{h(c)}}$, given that $h$ is a function such that $h(c)\neq 0$
2. Given a continuous function $f$ and $g$ over any interval $I$, prove that $f\circ \lim _{x\rightarrow a}g(x)=\lim _{x\rightarrow a}f\circ g(x)$ for all $x$ in the interval $I$

Comments and Further Reading

Question 2 is the proof that a limit can "transfer" between a composition of functions.

Unsorted 2

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 $|f'(x)|\leq K$ for all $x\in (a,b)$.
   1. Prove that $f$ is Lipschitz continuous on $(a,b)$
   2. Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function $f$ you are working with is uniformly continuous).

Hint (Question 6.1)

The Mean Value Theorem can be used here.

Hints/Answers

1. No hint.
2. You may want to prove first that the region above a convex function is convex (i.e. any straight line joining two points in the region, lies wholly in the region) and then using this fact argue by way of contradiction to show that convex functions are indeed continuous (i.e. no jump or removable discontinuity)
3. Consider the function $h(x)=f(x)-x$. Using the Intermediate Value Property, show that $\exists p$ such that $h(p)=0$.
4. First show that the set of all infinite sequences of real numbers has the same cardinality as $\mathbb {R}$ and next show that every continuous function is determined by its values on $\mathbb {Q}$
5. No hint.
6. (a) Use mean value theorem, once we cover it. (b) Let $\delta =\epsilon /K$.