This is an interesting take on a rigorous calculus, aimed at Honors Calculus students, although I think it takes on too many different tasks for its intended audience. I have no idea what "deconstructing calculus" means, but you can skip over that.
The book assumes no previous experience with proofs. It weaves the techniques of mathematical reasoning into the exposition as needed, and has an appendix on "Mathematical Rhetoric" that gives a more concentrated version. I think this approach works well.
The book, although rigorous, is not complete (it does note the omissions in footnotes). For example, the treatment of implicit differentiation assumes that the implicitly-defined function is differentiable (p. 181). There's no construction of the real numbers, and most of their properties are taken for granted (completeness is stated as an axiom on p. 27).
The book's strength is that it has very challenging exercises, even in the early sections. The body of the book usually covers only the most vital portions of calculus, with many more advanced topics sketched in the exercises.
The book's weakness is that there's too much going on. It's often hard to tell where we are going. In addition to a rigorous development of calculus, there are a lot of cookbook sections on problem-solving techniques, and quite a lot of detailed discussion of the history of calculus and earlier methods used before the modern streamlined subject was developed.
This is not a bad book, but there are better books. I think a better choice for a rigorous calculus at this level is Michael Spivak's Calculus (whose 4th edition came out in 2008). Spivak's book is more focused than Nitecki's, and does a better job of explaining why proof is important. As references I like G. H. Hardy's Pure Mathematics and Edmund Landau's Differential and Integral Calculus. By present-day standards these are not textbooks, and I would not recommend them for any course, but they do have concise, clear proofs of any calculus fact you need.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.