Some questions about Banach's proof of the existence of continuous nowhere differentiable functions Im reading Continuity and Category of Chapter11, Carothers' Real Analysis, 1ed. Here is a reading material at the end of this chapter, talking about Banach's proof of the existence of continuous nowhere differentiable functions,
I have 4 questions here,


*

*I cannot understand the sentence in paragraph3 that is " In particular, any f∈$C$[0,1] having a right-hand derivative at most n in magnitude at even one point in [0, 1-(1/n)] is in $E_n$". I mean what is "at most n in magnitude "? Im not an english native speaker.

*How did he guarantee that |f(x+h)-f(x)| =< nh for all 0 < h < 1-x ?

*Why does the proof merely focus on right-hand derivatives?

*What is the quotients involved in?
 A: *

*"at most $n$ in magnitude" means "of absolute value $\leq n$".

*If $f$ has a right hand derivative at some point $x$ whose
absolute value $< n$, then $|f(x+h) - f(x)| \leq n h$ if $h$ is small
enough, say $0 < h < epsilon$.  On the other hand, if $\epsilon > 0$, then
since $f$ is continuous, we can find $n' > 0$ so that $|f(x + h) - f(x)| \leq n'h$
for $\epsilon \leq h \leq 1 - x$ (just because $| f(x+h) - f(x)|$ is bounded on
the interval $\epsilon \leq h \leq 1 - x$, as $f$ is continuous, and $h$ is bounded below by $\epsilon$). So, letting $n'' = \max\{n,n'\}$, we find that if $f$ has a right hand derivative $<n$ at some point, then $f \in E_{n''}$.
(The authors actually asserts this with $n'' = n$ and with $\leq n$ rather than $< n$ as the condition; I don't see this right now, but I could well be missing something, and it doesn't matter anyway; in the end all we care about is that
$f \in E$.)

*Focusing on right-hand derivatives gives an even stronger result (differentiable means that the two-sided derivative  exists, and this implies that the right-hand derivatie exists).  Presumably it also makes the analsis of 
the set $E$ a bit easier, because the set $E_n$ is simpler to define than if
we tried to write down an analogous set that fited well with two-sided derivatives.

*"difference quotient" mean the expression $
\bigl(f(x+h) - f(x)\bigr)/h$.  "Right hand difference quotient" means we're
taking $h > 0$.  If $f$ is in $E_n$ then the abs. value of this is bounded by $n$, just by definition of $E_n$.  So $f \in E$ if and only if it has bounded r.h.d.q.'s at some point $x \in [0,1).$ (Just take $n$ to be $\geq $ whatever the bound on the abs. value is.) 
