Real Analysis/Limits and Continuity Exercises

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Real Analysis Exercises

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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.