We consider the stochastic partial differential equation (SPDE) $$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$$ ∂ t u = 1 2 ∂ x 2 u + b ( u ) + σ ( u ) W ˙ , where $$u=u(t,x)$$ u = u ( t , x ) is defined for $$(t,x)\in (0,\infty )\times \mathbb {R}$$ ( t , x ) ∈ ( 0 , ∞ ) × R and $$\dot{W}$$ W ˙ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u (0) is bounded and measurable, and b and $$\sigma $$ σ are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.