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Difference delta operator can be written without a limit:

$$\Delta[f(x)]=f(x+1)-f(x)$$

The same is true for any other finite difference operator. But what about derivative? Is there a proof that it cannot be expressed similarly without a limit?

Note that I know that derivative can be expressed without a limit if reals are extended to hyperreals, for instance, and with other extensions. But I am interested with standard reals.

Anixx
  • 9,119
  • I am failing to see the motivation. What is the problem with limits exactly? It would be hard to do much analysis at all without them. – Math1000 Apr 06 '15 at 01:18
  • I would say no because just changing one sequence of points is enough to make a function change from having a limit to having no limit at a point, and there is no way to distinguish them without quantifiers unless you use other notation like "$\to$". – user21820 Apr 06 '15 at 01:19

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