  This paper introduces the continuous tensor abstraction, allowing indices to
take real-number values (e.g., A[3.14]), and provides a continuous loop
construct that iterates over the infinitely large set of real numbers. This
paper expands the existing tensor abstraction to include continuous tensors
that exhibit a piecewise-constant property, enabling the transformation of an
infinite amount of computation into a finite amount. Additionally, we present a
new tensor format abstraction for storing continuous tensors and a code
generation technique that automatically generates kernels for the continuous
tensor abstraction. Our approach introduces a novel method for loop-level
reasoning in domains like computational geometry and computer graphics,
traditionally unexplored in tensor programming models. Our approach
demonstrates comparable performance to hand-optimized kernels in leading
libraries across diverse applications. Compared to hand-implemented libraries
on a CPU, our compiler-based implementation achieves an average speedup of
9.20x on 2D radius search with ~100x fewer lines of code (LoC), 1.22x on
genomic interval overlapping queries (with ~26x LoC saving), and 1.69x on
trilinear interpolation in Neural Radiance Field (with ~9x LoC saving).
