Can calculus be used for real number? I recently learned that there is no positive infinitesimal real number. The only infinitesimal real number is 0.
For calculus integral, dx is always interpreted as infinitesimal small but non-zero which contradict with the property of real number.
And I don't think zero would be a valid infinitesimal number for calculus as real numbers divided by zero is undefined.
So my questions are:
1) Can we use calculus for real number in strictly speaking?
2) Is it correct to say the infinitesimal concept for calculus doesn't exist under real number system?

The standard calculus track includes many tricks for circumventing the need for actual infinitesimals.
For example, one of the main principles at work is that sufficiently small real numbers behave like infinitesimals. So, if you consider real numbers at all possible scales, you ensure that your argument involves sufficiently small scales — thus the $\epsilon$-$\delta$ argument that is so important for doing low-level arguments in calculus.
Another is to rescale quantities so that they aren't infinitesimal anymore. If $y$ is a function of $x$, while one might ostensibly be more interested in the quantities $\Delta y$ and $\Delta x$, we instead consider their ratio $\frac{\Delta y}{\Delta x}$ which 'cancels' out the infinitesimal scale to produce quantities large enough that we can study in terms of the standard reals; thus the definition of the derivative
$$ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$
You see these two basic themes:


*

*Argue in terms of the 'sufficiently small' or 'suffiicently large'

*Adjust the notions you use so as to cancel out infinitesimal or infinite scales


all throughout standard analysis.
