Introduction:
When the density of a rod increases continuously from one end to the other, it is important to consider the clamping and application of perpendicular force to ensure stability and prevent deformation. The clamping side and the direction of the force depend on the distribution of density along the rod.

Factors to consider:
1. Density gradient: The rate at which the density of the rod increases from one end to the other determines the clamping side and the force direction.
2. Material properties: The material of the rod, its strength, and elasticity need to be considered to prevent buckling or breaking.

Determining the clamping side:
To determine the clamping side, we need to analyze the density gradient along the rod. If the density increases from end A to end B, it is recommended to clamp the rod at end A.

Reasoning:
Clamping the rod at the end with lower density (end A) provides stability and prevents deformation or bending. When the rod is clamped at the end with higher density (end B), the density gradient can cause uneven expansion and contraction, leading to potential damage or failure.

Applying perpendicular force:
After determining the clamping side, the direction of the perpendicular force needs to be considered. The force should be applied perpendicular to the rod's length, exerting pressure towards the clamped end (end A).

Reasoning:
Applying the force towards the clamped end balances the density gradient. The force counteracts the higher density at end B, preventing the rod from bending or deforming due to the density variation.

Conclusion:
When the density of a rod increases continuously from end A to end B, it is recommended to clamp the rod at end A and apply the perpendicular force towards the clamped end. This arrangement ensures stability, prevents deformation, and balances the density gradient along the rod.