discrete-points-below-asymptote

Discrete Points Below the Asymptote

Summary: §§515, 517. A parity anomaly that produces, in transcendental curves like $y=a{e}^{x/b}$ and $y=(-1{)}^x$, infinitely many isolated points dense on lines parallel to the axis but forming no continuous branch. “A paradox which never occurs in algebraic curves.”

Sources: chapter21 §§515, 517.

Last updated: 2026-05-12

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The phenomenon for the logarithmic curve (§515)

The continuous branch $MBm$ of the logarithmic-curve $y=a{e}^{x/b}$ stays above the asymptote (positive $y$). But:

• If $x/b$ is an integer or a fraction with odd denominator, $y$ has a single positive value — no anomaly.
• If $x/b$ is a fraction with even denominator, $y=a{m}^{x/b}$ involves a root with even index, hence two values: one positive (on the main branch) and one negative.

The negative values lie below the asymptote. They occur exactly at even-denominator fractions in $x/b$ — a dense but countable subset of any interval. Between any two abscissas there are infinitely many even-denominator fractions, hence infinitely many such points; yet no two of them are contiguous (irrationals and odd-denominator fractions interleave). The points form no continuous branch.

An infinite number of such discrete points are below the asymptote, but they do not form a continuous branch of the curve, although infinitely small intervals constitute a continuous curve. This is a paradox which never occurs in algebraic curves. (source: chapter21, §515)

So the logarithmic curve technically has a “shadow” of isolated points on the opposite side of the asymptote, totally invisible to plotting but logically required by the equation.

The phenomenon for $y=(-1{)}^x$ (§517)

Even more stark: $y=(-1{)}^x$.

• $x$ an even integer or a fraction with even numerator (and odd denominator, e.g. $2/3$): $y=1$ — point at unit distance above the axis.
• $x$ an odd integer or a fraction with both numerator and denominator odd: $y=-1$ — point at unit distance below.
• Otherwise (fraction with even denominator, or irrational): $y$ is complex — no real point.

Between any two different values for the abscissa, no matter how close they may be, there is not only one, but an infinite number of different fractions with odd denominators. To each of these fractions there corresponds a point belonging to the proposed equation. (source: chapter21, §517)

The entire curve $y=(-1{)}^x$ is just two horizontal lines $y=+1$ and $y=-1$ filled with discrete points, dense but never contiguous. The same phenomenon spreads to any equation $y=(-a{)}^x$ where a negative is raised to a variable power.

Why algebraic curves cannot show this

In an algebraic curve $F(x,y)=0$, fixing $x$ leaves a polynomial $F(x,y)=0$ in $y$, with finitely many roots. As $x$ varies continuously, these roots vary continuously (away from singular points). One cannot have “infinitely many extra points dense on a line” without an entire branch of the curve coinciding with that line — in which case it is a line, not a scatter of points.

The phenomenon is therefore a signature of transcendence: a feature only available to multi-valued transcendental relations.

Connection to the infinite-logarithms paradox (§516)

The §515 anomaly is the geometric face of §516’s algebraic paradox. The negative root that produces a point below the asymptote corresponds to one of the complex logarithms of a positive number being made real by even-denominator root extraction. See infinite-logarithms-paradox for the algebraic side.

Connection to the exponential curve $y=x^x$

The §518 exponential-curves discussion notes the same anomaly: $y=x^x$ has the continuous branch $MDB$ on the positive axis, but for negative $x$ only discrete points (with the same even-denominator rule). The continuous part “terminates abruptly at $B$”, which the law of continuity (continuous-and-discontinuous-curves) suggests cannot happen — except by virtue of the discrete-point shadow.

Related pages

• chapter-21-on-transcendental-curves
• logarithmic-curve — the canonical host of this anomaly.
• infinite-logarithms-paradox — algebraic root cause.
• exponential-curves — same anomaly on $y=x^x$.
• continuous-and-discontinuous-curves — Euler’s notion of continuity strained here.