One reason calc books are so weird:

consider integrals:
Newton etc. (18th? 17th? century) had an intuitive understanding of them;
19th century (Riemann) actually defined them;
20th century (Lebesgue) abstracted more, showed it led to measure etc.

...and calc books -give- the 19th century definition,
but the proceed as if they were in the 18th? century, just using them
as anti-derivatives.

Ditto for sequences/series, kinda:
they give 19th century definitions, but only have/expect 18th century
sophistication

(and they don't look at interesting subtleties like Abel sums
(e.g., 1 - 1 + 1 - 1 + 1 ... = 1/2, if you think of it as
lim_{x \to -1} p(x)
where p(x) is the power series for 1/(1-x) about 0, namely
$p(x)= 1 + x + x^2 + \dots + x^n + \dotsp$
)

ditto for functions and continuity
(for them a function is something you compute with, an elementary
(more properly, EL) function)
ditto continuity: give defn at a point, but use intuition...

(****** grrrrrrr *******
 continuous functions are -much- more subtle than they appear;
 look at nowhere differentiable functions, peano curves, fractals!)