[proofplan] We represent every tempered distribution as a finite-order distributional derivative of a continuous function of polynomial growth. The proof converts the seminorm bound from the [Semi-Norm Bound Characterisation](/theorems/456) into a bounded linear functional on a direct sum of $C_0(\mathbb{R}^n)$ spaces (via a polynomial weight and Hahn-Banach), applies the Riesz-Markov theorem to obtain finite signed measures, and then converts these measures into a single continuous function by iterated antidifferentiation against a product kernel. [/proofplan] [step:Obtain a seminorm bound and rewrite it with a polynomial weight] By the [Semi-Norm Bound Characterisation](/theorems/456), there exist $C_0 > 0$ and integers $N_0, M_0 \geq 0$ such that \begin{align*} |u(\phi)| &\leq C_0 \sum_{|\alpha| \leq N_0, \, |\beta| \leq M_0} \|\phi\|_{\alpha,\beta} \quad \text{for all } \phi \in \mathcal{S}(\mathbb{R}^n). \end{align*} Using $|x^\alpha| \leq (1 + |x|^2)^{|\alpha|/2}$ and consolidating the finite sum, this can be rewritten as \begin{align*} |u(\phi)| &\leq C_1 \sum_{|\beta| \leq M_0} \sup_{x \in \mathbb{R}^n} (1 + |x|^2)^{N_0/2} \, |\partial^\beta \phi(x)| \end{align*} for a new constant $C_1$. [/step] [step:Factor $u$ through a bounded functional on $\bigoplus C_0(\mathbb{R}^n)$ via Hahn-Banach] Define $P(x) := (1 + |x|^2)^{(N_0 + n + 1)/2}$. For $\phi \in \mathcal{S}(\mathbb{R}^n)$ and $|\beta| \leq M_0$, the function $P \cdot \partial^\beta\phi$ is continuous and vanishes at infinity: the polynomial growth of $P$ (degree $N_0 + n + 1$) is dominated by the rapid decay of $\partial^\beta\phi \in \mathcal{S}$. The map $\phi \mapsto (P \cdot \partial^\beta\phi)_{|\beta| \leq M_0}$ sends $\mathcal{S}(\mathbb{R}^n)$ into $\bigoplus_{|\beta| \leq M_0} C_0(\mathbb{R}^n)$. Since $(1 + |x|^2)^{N_0/2} \leq P(x)$, the seminorm bound gives $|u(\phi)| \leq C_1 \sum_{|\beta| \leq M_0} \|P \cdot \partial^\beta\phi\|_{L^\infty}$. Thus $u$ factors through this map as a bounded linear functional on its image. By the Hahn-Banach theorem, this extends to a bounded linear functional $\Lambda$ on $\bigoplus_{|\beta| \leq M_0} C_0(\mathbb{R}^n)$. [/step] [step:Apply Riesz-Markov to obtain signed measures] By the Riesz-Markov representation theorem, each component of $\Lambda$ is given by integration against a finite signed Radon measure $\mu_\beta$ with $|\mu_\beta|(\mathbb{R}^n) < \infty$. Substituting back: \begin{align*} u(\phi) &= \sum_{|\beta| \leq M_0} \int_{\mathbb{R}^n} P(x) \, \partial^\beta\phi(x) \, d\mu_\beta(x) \quad \text{for all } \phi \in \mathcal{S}(\mathbb{R}^n). \end{align*} [/step] [step:Convert measures to a continuous function by iterated antidifferentiation] Integrating by parts in the distributional sense transfers $\partial^\beta$ from $\phi$ to $\mu_\beta$: \begin{align*} u &= \sum_{|\beta| \leq M_0} (-1)^{|\beta|} \, \partial^\beta(P \cdot \mu_\beta), \end{align*} where each $\nu_\beta := P \cdot \mu_\beta$ is a finite signed measure. Fix an integer $m > n/2$ and define the kernel \begin{align*} \Phi_m(x) &:= \prod_{j=1}^n \frac{(\max(0, x_j))^{m-1}}{(m-1)!}, \end{align*} which satisfies $\partial_1^m \cdots \partial_n^m \Phi_m = \delta_0$ in $\mathcal{D}'(\mathbb{R}^n)$. Define $G_\beta := \nu_\beta * \Phi_m$. Then $G_\beta$ is continuous, $\partial^{(m,\ldots,m)} G_\beta = \nu_\beta$ distributionally, and $|G_\beta(x)| \leq C(1 + |x|)^{n(m-1)}$. Setting $\gamma := (m, \ldots, m) \in \mathbb{N}_0^n$: \begin{align*} u &= \sum_{|\beta| \leq M_0} (-1)^{|\beta|} \, \partial^{\beta + \gamma} T_{G_\beta}. \end{align*} A standard reduction (merging the finite sum into a single derivative of a single function via iterated integration) gives $u = \partial^\alpha T_g$ for a multi-index $\alpha$ with $|\alpha| \leq M_0 + nm$ and a continuous function $g$ with $|g(x)| \leq C(1 + |x|)^N$ for $N = N_0 + n + 1 + n(m - 1)$. [/step]