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Everything I've encountered in physics so far allows infinitesimal numbers to be manipulated as real numbers. But there has been much criticism towards Leibniz's notation, and I assume it is for a reason. When in mathematics will the infinitesimal notation not work? Including treating [itex]\frac{dy}{dx}[/itex] as a fraction to solve differential equations and such. Does it ever breakdown? If so, are those purely mathematical, scholarly problems, or is there places where the notation won't work in physics?