Calculus via Infinitesimals: Continuity and Differentiation

Where the payoff begins. This page of the NSA spine rebuilds the central notions of one-variable calculus — continuity and the derivative — directly with infinitesimals, and proves each nonstandard definition equivalent to its classical $\epsilon$–$\delta$ counterpart (continuity.md, differentiation.md). The result is that Leibniz's $\frac{dy}{dx}$ is, after all, a genuine quotient — of hyperreals — whose standard part (hyperreals.md §4) is the derivative.

References: Keisler, Elementary Calculus, ch. 2–3; Goldblatt, Lectures on the Hyperreals, ch. 7–8; Robinson, Non-standard Analysis, ch. 4.

1. Microcontinuity

Let $f:\mathbb{R}\to \mathbb{R}$ (or on a real interval) and $a\in \mathbb{R}$. Recall $x\approx y$ means $x-y$ is infinitesimal (hyperreals.md).

Definition (microcontinuity). $f$ is microcontinuous at $a$ if $$x\approx a⟹{}^{\ast }f(x)\approx {}^{\ast }f(a)>\text{for all }x\in {}^{\ast }\mathbb{R}.$$

In words: whenever the input is infinitely close to $a$, the output is infinitely close to $f(a)$. Compare the leaden $\epsilon$–$\delta$ phrasing — microcontinuity is a single implication between halos (hyperreals.md §3): ${}^{\ast }f$ maps $\mathrm{hal}(a)$ into $\mathrm{hal}(f(a))$.

Theorem (equivalence). $f$ is continuous at $a$ (in the $\epsilon$–$\delta$ sense) iff $f$ is microcontinuous at $a$.

Proof. ($\Rightarrow$) Assume continuity and let $x\approx a$. Fix real $\epsilon >0$; continuity gives real $\delta >0$ with $|t-a|<\delta \Rightarrow |f(t)-f(a)|<\epsilon$ for all real $t$. Transfer this sentence: $|x-a|<\delta \Rightarrow |{}^{\ast }f(x)-f(a)|<\epsilon$ for all hyperreal $x$ (transfer-principle.md). Since $x\approx a$, $|x-a|<\delta$, so $|{}^{\ast }f(x)-f(a)|<\epsilon$. As $\epsilon$ was an arbitrary real, ${}^{\ast }f(x)\approx f(a)$. ($\Leftarrow$) Contrapositive: if continuity fails, some real $\epsilon >0$ has, for every real $\delta >0$, a point $t_{\delta }$ with $|t_{\delta }-a|<\delta$ but $|f(t_{\delta })-f(a)|\ge \epsilon$. Taking $\delta =\frac{1}{n}$ gives a sequence $t_n\to a$ with $|f(t_n)-f(a)|\ge \epsilon$; its ultrapower class $x=[t_n]$ satisfies $x\approx a$ yet $|{}^{\ast }f(x)-f(a)|\ge \epsilon$, so microcontinuity fails. $\square$

2. Uniform continuity is "microcontinuity at every hyperreal"

The classical distinction between pointwise and uniform continuity (continuity.md §5) becomes a clean quantifier move over ${}^{\ast }\mathbb{R}$:

Theorem. $f$ is uniformly continuous on a set $E$ iff $$x\approx y⟹{}^{\ast }f(x)\approx {}^{\ast }f(y)>\text{for all }x,y\in {}^{\ast }E.$$

The difference from §1 is that $x,y$ range over all of ${}^{\ast }E$ (including nonstandard points), not just halos of standard points. This instantly explains the standard examples: $f(x)=x^2$ fails on $\mathbb{R}$ because for infinite $\omega$, $\omega \approx \omega +\frac{1}{\omega }$ but $(\omega +\frac{1}{\omega }{)}^2-{\omega }^2=2+\frac{1}{{\omega }^2}\not\approx 0$. And it makes Heine–Cantor a one-liner: on a compact $E$ every point of ${}^{\ast }E$ is near-standard (sequences-topology.md), so pointwise microcontinuity upgrades to the uniform version.

3. The derivative as a standard part

Definition (derivative). $f$ is differentiable at $a$ with derivative $f^{\prime }(a)=L\in \mathbb{R}$ if for every nonzero infinitesimal $\Delta x$, $$\frac{{}^{\ast }f(a+\Delta x)-f(a)}{\Delta x}\approx L,>\text{i.e.}>f^{\prime }(a)=\mathrm{st}\left(\frac{{}^{\ast }f(a+\Delta x)-f(a)}{\Delta x}\right),$$ the same value $L$ for all such $\Delta x$.

The quotient is an honest division of hyperreals — legal because $\Delta x\ne 0$ (hyperreals.md §1) — and $f^{\prime }(a)$ is its shadow. The increment $\Delta y={}^{\ast }f(a+\Delta x)-f(a)$ is infinitesimal exactly when $f$ is microcontinuous, so differentiable $\Rightarrow$ continuous is immediate: $\Delta y=(L+\eta )\Delta x\approx 0$ for infinitesimal $\eta$.

Equivalence with the limit definition. $\mathrm{st}\left(\frac{{}^{\ast }>f(a+\Delta x)-f(a)}{\Delta x}\right)=L$ for all infinitesimal $\Delta x\ne 0$ iff ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}=L$. The proof is the microcontinuity argument of §1 applied to the difference-quotient function — a transfer computation. So the two derivatives coincide wherever either exists (differentiation.md).

3.1 Worked example

For $f(x)=x^2$ at general $a$, with $\Delta x$ any nonzero infinitesimal:

$$\frac{(a+\Delta x{)}^2-a^2}{\Delta x}=\frac{2a\Delta x+(\Delta x{)}^2}{\Delta x}=2a+\Delta x\approx 2a,\text{so}{f}^{\prime }(a)=\mathrm{st}(2a+\Delta x)=2a.$$

The "$+\Delta x$" that a physicist drops is discarded rigorously by $\mathrm{st}$ — and the term $(\Delta x{)}^2$ never had to be "neglected," it simply canceled. This is the exact resolution of Berkeley's paradox promised in motivation-history.md §3.

4. The differentiation rules, infinitesimally

Every rule becomes elementary hyperreal algebra followed by $\mathrm{st}$; the standard-part laws (hyperreals.md §4.1) do the bookkeeping. Write $\Delta x\ne 0$ infinitesimal, $\Delta f={}^{\ast }f(a+\Delta x)-f(a)$, and recall $\Delta f=(f^{\prime }(a)+\eta )\Delta x$ with $\eta \approx 0$.

• Product rule. $\Delta (fg)=f\Delta g+g\Delta f+\Delta f\Delta g$; divide by $\Delta x$ and take $\mathrm{st}$ — the cross term $\frac{\Delta f\Delta g}{\Delta x}=\Delta f\cdot \frac{\Delta g}{\Delta x}\approx 0\cdot g^{\prime }(a)=0$ vanishes, leaving $f{g}^{\prime }+g{f}^{\prime }$.
• Chain rule is now literal fraction cancellation: with $u={}^{\ast }g(a+\Delta x)$ and $\Delta u=u-g(a)$ (infinitesimal, nonzero for the generic case), $$\frac{\Delta (f\circ g)}{\Delta x}=\frac{\Delta f}{\Delta u}\cdot \frac{\Delta u}{\Delta x}\approx f^{\prime }(g(a))g^{\prime }(a).$$ (The degenerate case $\Delta u=0$ is handled by the standard $\eta$-trick.) This is precisely the physicist's "cancel the $du$" — sound because $\Delta u$ is a genuine nonzero infinitesimal.
• Quotient rule follows the same pattern from $\Delta (f/g)$.

5. Existence theorems via near-standardness

The intermediate and extreme value theorems (continuity.md §3–4) get transparent nonstandard proofs; the recurring device is: take a hyperfinite grid (internal-sets.md §4), find the winning node by transfer of a finite fact, then take its standard part.

Extreme Value Theorem (nonstandard proof). Let $f$ be continuous on $[a,b]$. Partition into $H$ equal pieces ($H$ unlimited), nodes $x_i=a+i\frac{b-a}{H}$. By transfer of "a finite list has a greatest element," some node $x_M$ maximizes ${}^{\ast }f$ over the (hyperfinite) grid. Let $c=\mathrm{st}(x_M)\in [a,b]$. For any $x\in [a,b]$, some grid node $x_i\approx x$, and microcontinuity gives ${}^{\ast }f(x)\approx {}^{\ast }f(x_i)\le {}^{\ast }f(x_M)\approx f(c)$; taking standard parts, $f(x)\le f(c)$. So $f$ attains its max at $c$. $\square$

The intermediate value theorem is analogous: on the grid pick the last node with ${}^{\ast }f(x_i)<y$; its shadow is the desired root.

Where this page is used

• The derivative-as-standard-part and microcontinuity feed directly into nonstandard-integration.md (via the FTC) and sequences-topology.md.
• Provides the infinitesimal proofs that vindicate physics practice (physics-use-of-calculus.md §4.1, applications.md).
• Every equivalence is a transfer argument (transfer-principle.md).

Next: Non-Standard Integration — the integral as a hyperfinite sum of infinitesimal slices.