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  "body": "\n## Summary\n\nA continuous kernel is a [[kernel|GSMBXT7FPK]] $K(x, s)$ that is a continuous function of both variables on a closed bounded square $[a, b] \\times [a, b]$.\nIt is the setting of the classical theory of Fredholm and Volterra.\nThe integral operator with a continuous kernel maps continuous functions to continuous functions and is a [[compact operator|M2NEP1K5AE]] on the space of continuous functions.\nIteration methods, the Liouville–Neumann series and the Fredholm determinant are first developed for such kernels.\n\n## Explanations\n\n### L1 human\n\nA continuous kernel has no jumps and no infinite values.\nSince the square is closed and bounded, the kernel is bounded by some number $M$, which makes many estimates easy.\nFor example, $K(x, s) = e^{x s}$ and $K(x, s) = \\min(x, s)$ are continuous kernels on $[0, 1] \\times [0, 1]$.\nThe kernel $1/\\abs{x - s}$ is not, because it is infinite on the diagonal.\n",
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