Classical counterparts for the trigonometry-free oscillators

The Takagi series of The Takagi series converges uniformly to a continuous nowhere differentiable function and the classical Weierstrass function of Under $ab>1+3\pi \mathrm{/}2$, the classical Weierstrass function is continuous everywhere and differentiable nowhere have the same regularity verdict, but their mechanisms differ: the Takagi proof uses dyadic affine slopes, while the Weierstrass proof uses trigonometric probes and a one-signed frequency tail.

The distance-to-the-integers oscillator is a trigonometry-free Lipschitz model; the classical estimates that replace it are the sine and cosine inequalities in Sine and cosine are $1$-Lipschitz on $\mathbb{R}$. For reciprocal oscillation, sin(1/x) has no limit as x tends to zero, x sin(1/x) tends to zero despite its oscillation, The extension of x^2 sin(1/x) by zero is differentiable but its derivative is discontinuous at zero, and $x^2\sin (1\mathrm{/}{x}^2)$ has an unbounded, non-Riemann-integrable derivative ↗ record the undamped, once-damped, and twice-damped classical forms. The damping controls the value at zero, but differentiating can restore an oscillatory or unbounded term.