This paper introduces continuous tensors, an abstraction where indices can be real numbers (e.g., A[3.14]), extending traditional tensor algebra to continuous domains. By representing these as piecewise-constant functions, the system enables infinite-domain processing in finite time, allowing fields like computer graphics, genomics, and computational geometry to be expressed purely as tensor programs. The authors’ compiler-based implementation achieves significant performance gains—such as a 9.2x speedup on 2D radius searches—while reducing the required lines of code by up to 60x compared to hand-optimized kernels in leading libraries.