This paper is concerned with inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump. We nest previous works that assume either continuity or discontinuity at the threshold point and develop robust inference methods on the parameters of the model, which are valid under both specifications. In particular, we found that the parameter values under the kink restriction are irregular points of the Hessian matrix of the expected Gaussian quasi-likelihood. This irregularity destroys the asymptotic normality and induces the non-standard cube root convergence rate for the threshold estimate. However, it also enables us to obtain the same asymptotic distribution as in Hansen (2000) for the quasi-likelihood ratio statistic for the unknown threshold up to an unknown scale parameter. We show that this scale parameter can be consistently estimated by a kernel method as long as no higher order kernel is used. Furthermore, we propose to construct confidence intervals for the unknown threshold by bootstrap test inversion, also known as grid bootstrap. Finite sample performances of the grid bootstrap confidence intervals are examined through Monte Carlo simulations. We also implement our procedure to an economic empirical application.