"""GIFs for the note on the epsilon-delta definition of a limit. usage: python make_gifs.py -> images/blog/epsilon-delta-limit-definition/*.gif Needs matplotlib + Pillow and the 'IBM Plex Sans KR' font. """ from pathlib import Path import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation, PillowWriter from matplotlib.patches import Rectangle ROOT = Path(__file__).resolve().parents[3] OUT = ROOT / "images" / "blog" / "epsilon-delta-limit-definition" FPS = 14 HOLD_END = 28 INK, GREY, LIGHT = "#222222", "#8a8a8a", "#c8c8c8" ACCENT, ACCENT2 = "#c8553d", "#2f6690" plt.rcParams.update({ "font.family": ["IBM Plex Sans KR", "DejaVu Sans"], "mathtext.fontset": "dejavusans", "axes.unicode_minus": False, "font.size": 12, "figure.facecolor": "white", "axes.facecolor": "white", "savefig.facecolor": "white", }) def ease(t): t = np.clip(t, 0.0, 1.0) return 0.5 - 0.5 * np.cos(np.pi * t) def timeline(*segments): frames = [] for name, n in segments: for i in range(n): frames.append((name, i / (n - 1) if n > 1 else 1.0)) return frames def style_axes(ax, xlim, ylim, xticks, yticks): ax.set_xlim(*xlim) ax.set_ylim(*ylim) ax.set_xticks(xticks) ax.set_yticks(yticks) ax.grid(True, color="#eeeeee", lw=0.8, zorder=0) for s in ax.spines.values(): s.set_visible(False) ax.tick_params(colors="#555555", length=0, labelsize=10) def save(anim, fig, name, nframes): OUT.mkdir(parents=True, exist_ok=True) path = OUT / name anim.save(path, writer=PillowWriter(fps=FPS)) plt.close(fig) print(f"{name}: {nframes} frames, {path.stat().st_size / 1024:.0f} KB") A, L = 3.0, 9.0 xs = np.linspace(0.8, 5.2, 600) # (a) the epsilon band shrinks; delta = min(1, eps/7) answers each time def gif_game(): fig, ax = plt.subplots(figsize=(7.2, 5.0), dpi=100) fig.subplots_adjust(left=0.08, right=0.97, bottom=0.08, top=0.82) style_axes(ax, (0.8, 5.2), (0, 26), range(1, 6), range(0, 27, 5)) ax.plot(xs, xs**2, color=INK, lw=2.2, zorder=3) ax.scatter([A], [L], s=46, facecolor="white", edgecolor=INK, lw=1.6, zorder=6) eps_band = ax.add_patch(Rectangle((0.8, L - 1), 4.4, 2, color=ACCENT2, alpha=0.12, zorder=1, lw=0)) del_band = ax.add_patch(Rectangle((A - 0.1, 0), 0.2, 26, color=ACCENT, alpha=0.12, zorder=1, lw=0)) eps_lines = [ax.axhline(0, color=ACCENT2, lw=1.0, ls="--", zorder=2) for _ in range(2)] del_lines = [ax.axvline(0, color=ACCENT, lw=1.0, ls="--", zorder=2) for _ in range(2)] inside, = ax.plot([], [], color=ACCENT, lw=4.0, alpha=0.85, zorder=4) eps_txt = ax.text(4.95, 0, "", color=ACCENT2, fontsize=11, ha="right", va="bottom") del_txt = ax.text(0, 0.6, "", color=ACCENT, fontsize=11, ha="center") title = fig.text(0.08, 0.93, r"$\lim_{x\to 3} x^2 = 9$: 상대가 $\varepsilon$을 내면 $\delta=\min(1,\varepsilon/7)$로 답한다", fontsize=13, color=INK, va="center") sub = fig.text(0.08, 0.865, "", fontsize=11.5, color="#555555", va="center") eps_path = [6.0, 3.0, 1.5, 0.6] segs = [("hold0", 14)] for i in range(1, len(eps_path)): segs += [(f"move{i}", 22), (f"hold{i}", 14)] segs += [("end", HOLD_END)] frames = timeline(*segs) def eps_at(name, t): if name.startswith("move"): i = int(name[4:]) return eps_path[i - 1] + (eps_path[i] - eps_path[i - 1]) * ease(t) if name.startswith("hold"): return eps_path[int(name[4:])] return eps_path[-1] def update(fr): name, t = fr e = eps_at(name, t) d = min(1.0, e / 7) eps_band.set_y(L - e) eps_band.set_height(2 * e) del_band.set_x(A - d) del_band.set_width(2 * d) for ln, y in zip(eps_lines, (L - e, L + e)): ln.set_ydata([y, y]) for ln, x in zip(del_lines, (A - d, A + d)): ln.set_xdata([x, x]) xi = np.linspace(A - d, A + d, 200) inside.set_data(xi, xi**2) eps_txt.set_text(f"9 ± {e:.2f}") eps_txt.set_position((5.15, L + e + 0.3)) del_txt.set_text(f"3 ± {d:.3f}") del_txt.set_position((A, 0.7)) worst = max((A + d) ** 2 - L, L - (A - d) ** 2) sub.set_text(f"ε = {e:.2f}, δ = {d:.3f}: 구간 안 최대 오차 {worst:.2f} < ε") return [] anim = FuncAnimation(fig, update, frames=frames, blit=False) save(anim, fig, "epsilon-delta-game.gif", len(frames)) # (b) eps = 10: delta = eps/7 lets x = 4.4 escape, delta = min(1, eps/7) does not def gif_counterexample(): fig, axes = plt.subplots(1, 2, figsize=(7.2, 4.6), dpi=100, sharey=True) fig.subplots_adjust(left=0.08, right=0.98, bottom=0.09, top=0.80, wspace=0.08) e = 10.0 deltas = [e / 7, min(1.0, e / 7)] labels = [r"$\delta=\varepsilon/7\approx1.43$", r"$\delta=\min(1,\varepsilon/7)=1$"] dots, vals = [], [] for ax, d, lab in zip(axes, deltas, labels): style_axes(ax, (1.2, 4.9), (0, 25), range(2, 5), range(0, 26, 5)) ax.axhspan(L - e, L + e, color=ACCENT2, alpha=0.12, zorder=1) ax.axhline(L + e, color=ACCENT2, lw=1.0, ls="--", zorder=2) ax.axvspan(A - d, A + d, color=ACCENT, alpha=0.12, zorder=1) ax.plot(xs, xs**2, color=INK, lw=2.2, zorder=3) ax.set_title(lab, fontsize=12.5, color=INK, pad=8) ax.text(1.3, L + e + 0.5, "9 + ε = 19", color=ACCENT2, fontsize=10.5) dots.append(ax.scatter([], [], s=50, zorder=6)) vals.append(ax.text(0.03, 0.04, "", transform=ax.transAxes, fontsize=11, va="bottom")) title = fig.text(0.08, 0.93, r"상대가 $\varepsilon=10$을 낸다. $x$를 구간 오른쪽 끝으로 밀어 본다", fontsize=13, color=INK, va="center") frames = timeline(("sweep", 46), ("end", HOLD_END)) def update(fr): name, t = fr s = ease(t) if name == "sweep" else 1.0 for i, (d, dot, v) in enumerate(zip(deltas, dots, vals)): x = A + s * (d * 0.98) if i == 0 and name == "end": x = 4.4 y = x * x bad = y > L + e dot.set_offsets([[x, y]]) dot.set_color(ACCENT if bad else ACCENT2) v.set_text(f"x = {x:.2f}, x² = {y:.2f}" + (" > 19" if bad else " < 19")) v.set_color(ACCENT if bad else ACCENT2) return [] anim = FuncAnimation(fig, update, frames=frames, blit=False) save(anim, fig, "min-counterexample.gif", len(frames)) if __name__ == "__main__": gif_game() gif_counterexample()