Time kink: modeling change of metric signature

The model of a signature change of a metric from the Lorenztian to Euclidean one with the use of a time dependent kink as $g_{00}$ component of the metric is considered. The metric which describes the continuous change of the signature of this type on a hypersurface is constructed and corresponding Einstein equations are solved in both regions of the space-time. The discontinuities of the Einstein tensor components on the hypersurface are discussed as well as junction conditions for the parameters of the solutions. Additionally, the properties of a transition from the space-time with one signature to an another are discussed, the presence of an inflation in the model is demonstrated as a consequence of the signature change without any additional fields required.