# Real Analysis/Limits and Continuity Exercises/Hints

 Real Analysis Exercises

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 ${\displaystyle f}$  and ${\displaystyle g}$ , prove the following
• ${\displaystyle \lim _{x\rightarrow c}{(f+g)(x)}=f(c)+g(c)}$
• ${\displaystyle \lim _{x\rightarrow c}{(f\cdot g)(x)}=f(c)\cdot g(c)}$
• ${\displaystyle \lim _{x\rightarrow c}{\left({\dfrac {f}{h}}\right)(x)}={\dfrac {f(c)}{h(c)}}}$ , given that ${\displaystyle h}$  is a function such that ${\displaystyle h(c)\neq 0}$
2. Given a continuous function ${\displaystyle f}$  and ${\displaystyle g}$  over any interval ${\displaystyle I}$ , prove that ${\displaystyle f\circ \lim _{x\rightarrow a}g(x)=\lim _{x\rightarrow a}f\circ g(x)}$  for all ${\displaystyle x}$  in the interval ${\displaystyle I}$

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 ${\displaystyle f:(a,b)\rightarrow \mathbb {R} }$  is a convex function if for all ${\displaystyle x,y\in (a,b)}$  and all ${\displaystyle s,t\in [0,1]}$  with ${\displaystyle s+t=1}$ , ${\displaystyle f(sx+ty)\leq sf(x)+tf(y)}$ )
3. Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is ${\displaystyle \exists p\in [0,1]}$  such that ${\displaystyle f(p)=p}$
4. Prove that the space of continuous functions on an interval has the cardinality of ${\displaystyle \mathbb {R} }$
5. Let ${\displaystyle f:[a,b]\rightarrow \mathbb {R} }$  be a monotone function, i.e. ${\displaystyle \forall x,y\in [a,b];x\leq y\Rightarrow f(x)\leq f(y)}$ . Prove that ${\displaystyle f}$  has countably many points of discontinuity.
6. Let ${\displaystyle f:(a,b)\rightarrow \mathbb {R} }$  be a differentiable function, and suppose there is some positive constant ${\displaystyle K}$  such that ${\displaystyle |f'(x)|\leq K}$  for all ${\displaystyle x\in (a,b)}$ .
1. Prove that ${\displaystyle f}$  is Lipschitz continuous on ${\displaystyle (a,b)}$
2. Show that every function which is Lipschitz continuous is also uniformly continuous (and therefore the function ${\displaystyle f}$  you are working with is uniformly continuous).
Hint (Question 6.1)

The Mean Value Theorem can be used here.

3. Consider the function ${\displaystyle h(x)=f(x)-x}$ . Using the Intermediate Value Property, show that ${\displaystyle \exists p}$  such that ${\displaystyle h(p)=0}$ .
4. First show that the set of all infinite sequences of real numbers has the same cardinality as ${\displaystyle \mathbb {R} }$  and next show that every continuous function is determined by its values on ${\displaystyle \mathbb {Q} }$
6. (a) Use mean value theorem, once we cover it. (b) Let ${\displaystyle \delta =\epsilon /K}$ .