It depends on what you want. As already discussed pure math has another aim than applied math. Pure math aims at rigorous proofs of theorems from a given set of axioms (or even investigates the implications of this idea itself in terms of formal logics). This is, of course, important also for the applications, because the proven theorems and the discovery of other theorems in proving provides the idea, how to apply mathematics to the description of the real world. The goal of the theoretical sciences is to find the right mathematical structures to describe some particular aspect of nature, e.g., the motion of bodies and continua (fluids and solid) in Newtonian or relativistic classical mechanics. For this you need to know the mathematical structures, found by the pure mathematicians, to find such a description. E.g., for Einstein when he looked for the right description of gravity within relativistic physics, he could build on the notion of Riemannian geometry, which was developed by pure mathematicians like Gauß, Lobatchevsky, Riemann, Minkowski, Levi-Civita et al. in the goal to investigate the status of the parallel axiom in Euclidean geometry, which is a completely pure-math academic question.
It's of course true that also the pure mathematicians need heuristic ideas before they build their abstract edifices. One famous example is Functional Analysis, which was developed to give unrigorous mathematical manipulations by physicists and engineers like the Dirac ##\delta## distribution (which was in fact invented much earlier by Sommerfeld) a solid foundation. From this Functional Analysis as a whole branch of mathematics developed.
That said, you must be clear about what's the goal you aim at before reading a math textbook: If you want to apply mathematics to real-world problems you need more something like applied calculus, where you learn the calculational techniques used in the theoretical sciences to describe and solve real-world problems. If you want to learn how to make the applied math rigorous and watertight, you should use an analysis textbook. The former kind of books won't teach you the rigorous math and the latter won't teach you how to calculate things in practice. I think, however, a good mathematician should also know a bit about how to calculate mundane things like an integral or solve a given differential equation as well as a good scientist should know a bit about the rigorous foundations of the math he/she applies in their daily work.