A smooth map can fail the tangential cone condition

Given: The standard real Hilbert line $X=Y=\mathbb{R}$, the map $F(t)=t^2$, basepoint $x^{†}=0$, exact datum $y=0$, and a proposed local TCC radius $\delta >0$ with constant $0\le \eta <1$.

The one-coordinate Euclidean inner product is $⟨s,t⟩=st$, with induced Euclidean norm (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

The induced Euclidean length on ${\mathbb{R}}^1$ is a norm (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

Every Cauchy sequence of real numbers converges (The reals are complete).

A real inner-product space complete in its induced norm is a real Hilbert space (Real and complex inner-product spaces and their induced length, Hilbert space).

A Fréchet derivative is a bounded linear map whose remainder divided by the increment norm tends to zero (Fréchet derivative between Banach spaces).

The local TCC at $0$ would require $\mathrm{\mid }F(h)-F(0)-F^{\mathrm{\prime }}(0)h\mathrm{\mid }\le \eta \mathrm{\mid }F(h)-F(0)\mathrm{\mid }$ whenever $\mathrm{\mid }h\mathrm{\mid }\le \delta$ (Local tangential cone condition).