\begin{piece}
{A Letter to the {\em Monthly} Editor}
{by Marvin Knopp}\footnote[1]{Editor's note: 
This is in response to Steven Krantz's review of Omar Hijab's
recent book,  which appears in \em Amer. Math. Monthly, \rm 105, no.
7 (1998), 677-682. This response was sent to the {\em Monthly}'s 
editor Roger A. Horn.}

Dear Professor Horn:
Steven Krantz's egregiously wrong-headed
review of Omar Hijab's recent book, ``Introduction to Calculus and
Classical Analysis'' (ICCA) [Amer. Math. Monthly 105, no.
7 (1998), 677-682], begins with a description of the
properties he feels a good mathematics textbook ought to
have.  At first appearing potentially useful, this turns
laughable in light of Krantz's own inability to recognize
a superb text (Hijab's) when he sees one.

Having used Hijab's book in a year-long course for
advanced undergraduates at Temple University during the
academic year just past, I feel qualified to comment both
upon the book and upon the reviewer's odd perception of
it. ICCA is beautifully conceived and carefully executed.
Understandably, therefore, I had a visceral reaction to
Krantz's bald and unsupportable assertion that ``the
entire text is sloppily written --- what I would think
of  as a first draft''. Not that Krantz makes no effort
to support his assertion. Indeed, as evidence he adduces
the author's 

\begin{itemize}
\item[(i)] failure to construct the real number system
$\bf R$,

\item[(ii)] relegation of the proof of uniqueness of
$\sup(S)$, \newline
$S\subset\bf R$, to a brief footnote.
\end{itemize}

Now, I like a good construction of $\bf R$ as much as
the next person, and in fact I discussed its importance
with my class, including as well a brief sketch of the
construction by way of Cauchy sequences of rationals. I
omitted the details for the same reason that the author
omits the construction: there is a good deal of serious
analysis to be covered in the remainder of the book,
especially in chapter 5, and the time is better spent in
the later material. Concerning the uniqueness of
$\sup(S)$ ---  here I devoted enough time amplifying
Hijab's terse (but logically complete) footnote to be
certain that all members of the class understood the
argument. 

The previous paragraph underscores a serious flaw in
Krantz's perspective: he appears to disregard entirely
the role of the instructor. He complains that the author
``echews $\epsilon$'s and $\delta$'s, \dots defines the
integral of a function to be the area under the graph,
almost completely eliminates uniform continuity and
uniform convergence \dots'' and says little of Riemann
sums and Riemann integrability. If there were no
instructor to guide the learning this would present a
problem (as would, indeed, any completely conventional
treatment).  But, my students did have an instructor, I
did succeed in exposing them to standard approaches to
the material along with Hijab's approach, and they
clearly benefited from the multiplicity of viewpoints. 

Honesty requires the disclosure here that most of the
credit for this pedagogical success belongs to ICCA. For
instance, a simple theorem on page 23 (\S1.5) relates
Hijab-style limits (upper and lower limits) and limits by
way of $\epsilon$-$\delta$.  Proving this theorem freed
me to present results on limits either way.  I made the
most of this freedom, presenting proofs in whichever mode
appeared simpler, and on occasion in both modes.

The author's unusual, but carefully thought out approach
to integration is even more liberating because of its
radical simplification of a complex subject. Defining the
integral as the area under the graph is a start towards
stripping away the complexity, but this is not nearly as
important as the fact (not noted by Krantz) that Hijab's
notion of area is outer area, rather than the traditional
Jordan content in ${\bf R}^2$. Initially, this was
worrisome, my traditional mindset warning me that this
was not going to work. To my surprise and delight, the
classroom experience showed that nothing could be further
from the truth.

For Riemann integrable functions (hence for continuous
functions) the ``Hijab integral'' coincides with the
Riemann integral, and it includes the Cauchy-Riemann
integral as well, at no additional charge. (In fact, for
Lebesgue measurable functions, the ICCA integral
coincides with  the Lebesgue integral. However,
measurable functions are neither needed nor discussed in
the text.) Uniform convergence plays no part in the
development, since Hijab presents instead a monotone
convergence theorem valid for arbitrary nonnegative
functions and a dominated convergence theorem for
continuous functions (Yes --- you read that right!).

Krantz closes his review of ICCA with praise for the
final chapter 5, calling it an ``astonishing {\it tour
de force}''. Here, at last, Krantz gets something right,
but though he is certain he ``would refer to [ICCA] for
ideas'' in his teaching, he remains doubtful he could
ever teach from it. This assertion notwithstanding, I am
certain Krantz could teach from the book and end up
enamored of it, as I was from the outset. He would
discover, as I did, that ICCA has much to teach, both
about mathematics and how to write mathematics. He would
find --- as I did when I assigned more than half of the
``astonishing'' (and difficult!) chapter 5 to my students
to read on their own and to present in class --- that
this book is eminently readable, even for ``tyros''. 

I urge Professor Krantz to teach from ICCA and then 
write the review he should have written in the first
place. For his efforts, he will be a wiser mathematician,
a happier teacher and a far better reviewer. Best of all
--- he won't have to read any more of my annoying letters
containing unwanted advice.

Sincerely, \\

 Marvin Knopp.

\end{piece}