Some questions about Banach's proof of the existence of continuous nowhere differentiable functions

Question

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,

1. 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.

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

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

4. What is the quotients involved in?

Answer 1

• "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.)