Real Analysis/Limits and Continuity 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 $f$ and $g$, prove the following
   • $\underset{x\to c}{\lim }(f+g)(x)=f(c)+g(c)$
   • $\underset{x\to c}{\lim }(f\cdot g)(x)=f(c)\cdot g(c)$
   • $\underset{x\to c}{\lim }\left(\frac{f}{h}\right)(x)=\frac{f(c)}{h(c)}$, given that $h$ is a function such that $h(c)\ne 0$
2. Given a continuous function $f$ and $g$ over any interval $I$, prove that $f\circ \underset{x\to a}{\lim }g(x)=\underset{x\to a}{\lim }f\circ g(x)$ for all $x$ in the interval $I$

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 $f:(a,b)\to ℝ$ is a convex function if for all $x,y\in (a,b)$ and all $s,t\in [0,1]$ with $s+t=1$, $f(sx+ty)\le sf(x)+tf(y)$)
3. Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is $\mathrm{\exists }p\in [0,1]$ such that $f(p)=p$
4. Prove that the space of continuous functions on an interval has the cardinality of $ℝ$
5. Let $f:[a,b]\to ℝ$ be a monotone function, i.e. $\mathrm{\forall }x,y\in [a,b];x\le y\Rightarrow f(x)\le f(y)$. Prove that $f$ has countably many points of discontinuity.
6. Let $f:(a,b)\to ℝ$ be a differentiable function, and suppose there is some positive constant $K$ such that $|f^{\prime }(x)|\le K$ 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).