= Solution
Let $p_t(z)=(2\pi t)^{-1/2}e^{-z^2/(2t)}$ be the <heat kernel> for $u_t=\frac12u_{xx}$. The symmetry of the <Gaussian distribution> gives the method-of-images formula
$$
v(t,x)=\int_0^\infty f(y)\bigl(p_t(x-y)+p_t(x+y)\bigr)dy.
$$
This is the <Neumann heat kernel on a half-line>. <Differentiation under the integral sign> shows that $v\in C^{1,2}$ for $t,x>0$ and that $v_t=\frac12v_{xx}$. At $x=0$, the two differentiated kernel terms cancel, so $v_x(t,0)=0$. The <Gaussian approximate identity> gives $v(t,x)\to f(x)$ as $t\downarrow0$, while the <dominated convergence theorem> gives continuity up to $x=0$. Finally $|v(t,x)|\leq\lVert f\rVert_\infty$, which is stronger than the required exponential bound. Thus $v$ satisfies every condition in the displayed boundary-value problem.
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