Stress Concentration at Structural Notch (Mechanical Engineering)

A mechanical engineer conducts a finite element stress simulation on a critical titanium aerospace load-bearing bracket. The boundary geometry along the inner tension surface of the bracket is described by the absolute-value profile:

where $x$ represents the longitudinal position in centimeters and $y(x)$ represents the surface contour.

1. Analytically prove that $y(x)$ is continuous but non-differentiable at the vertex $x = 5$.
2. Explain the physical consequence of this geometric non-differentiability regarding stress concentration factors ($K_t$) when the bracket is subjected to cyclic operational loads.
3. Propose the mathematical modification engineers implement (such as a circular or polynomial transition fillet) to restore differentiability and safeguard against fatigue crack initiation.