High-Speed Railway Transition (Civil Engineering)

A transportation civil engineer is designing a track layout for a new high-speed passenger railway. The horizontal position profile of the track alignment transitions from a straight access track into a curved bypass, modeled by the piecewise function:

where $x$ and $T(x)$ are measured in hundreds of meters.

1. Analytically determine if the track layout is continuous at the junction point $x = 3$.
2. Determine if the track layout is differentiable at $x = 3$ by comparing the left-hand and right-hand derivative limits.
3. Based on differentiability and lateral dynamics, explain why this transition geometry is either safe or hazardous for a passenger train traveling at high speeds.