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  "body": "\n## Counterexample\n\nOne could guess the following. A function of two variables is continuous when it is continuous in each variable for each fixed value of the other variable.\nThis is false.\nLet $f \\colon \\mathbb{R}^2 \\to \\mathbb{R}$ be given by $f(x, y) = \\dfrac{xy}{x^2 + y^2}$ for $(x, y) \\ne (0, 0)$ and $f(0, 0) = 0$.\nFor each fixed $b$, the function $x \\mapsto f(x, b)$ is continuous on $\\mathbb{R}$, and for each fixed $a$, the function $y \\mapsto f(a, y)$ is continuous on $\\mathbb{R}$.\nBut $f$ is not [[continuous|0JEJATBVD7]] at $(0, 0)$.\n\n## Remarks\n\nFor $b \\ne 0$, the function $x \\mapsto f(x, b)$ is a quotient of polynomials with a denominator that is never $0$. For $b = 0$, it is the constant $0$. The same holds for the other variable.\nOn the line $y = x$ with $x \\ne 0$, the value is $f(x, x) = 1/2$. So $f(x, x) \\to 1/2 \\ne f(0, 0)$ as $x \\to 0$, by the [[test with different paths|0BJ6H85DRQ]].\n\n## Explanations\n\n### L1 human\n\nSeparate continuity checks only the two directions along the axes.\nOn both axes this function is $0$, so nothing goes wrong there.\nOn the diagonal it is $1/2$ everywhere except at the origin, so the function jumps there.\n",
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  "name": "Separately continuous but not continuous function",
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