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  "body": "\n## Counterexample\n\nLet $f(x) = \\sin(1/x)$ for $x \\ne 0$.\nThen $\\lim_{x \\to 0} f(x)$ does not exist, and neither one-sided limit exists.\nLet $x_n = 1/(2 \\pi n + \\pi/2)$ and $y_n = 1/(\\pi n)$. Both sequences tend to $0$ from the right, but $f(x_n) = 1$ and $f(y_n) = 0$ for all $n \\ge 1$.\n\n## Remarks\n\nThe argument uses the [[sequential characterization of limits|000MXHEY22]].\nSo no value of $f$ at $0$ makes $f$ continuous at $0$. The point $0$ is an essential discontinuity.\nThe graph of $f$ on $(0, 1]$, with its closure, is the topologist's sine curve.\n\n## Explanations\n\n### L1 human\n\nAs $x$ approaches $0$, the number $1/x$ grows without bound.\nSo $\\sin(1/x)$ swings between $-1$ and $1$ faster and faster, and it settles at no value.\n",
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