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  2. Infinitesimal - Wikipedia

    en.wikipedia.org/wiki/Infinitesimal

    In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real number. Infinitesimals ...

  3. Nonstandard calculus - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_calculus

    Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

  4. Indeterminate form - Wikipedia

    en.wikipedia.org/wiki/Indeterminate_form

    Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.

  5. Nonstandard analysis - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_analysis

    A real-valued function f on the interval [a, b] is continuous if and only if for every hyperreal x in the interval *[a, b], we have: *f(x) ≅ *f(st(x)). Similarly, Theorem. A real-valued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h, the value

  6. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

  7. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.

  8. Increment theorem - Wikipedia

    en.wikipedia.org/wiki/Increment_theorem

    In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε , where Δ y = f ( x + Δ x ) − f ( x ...

  9. Leibniz's notation - Wikipedia

    en.wikipedia.org/wiki/Leibniz's_notation

    Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.. In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively ...