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  "body": "\n## Counterexample\n\nThe function $f(x) = |x|$ is continuous on $\\mathbb{R}$, but it is not differentiable at $0$.\nThe difference quotient is $\\frac{|h|}{h}$, which is $1$ for $h \u003e 0$ and $-1$ for $h \u003c 0$. So it has no limit as $h \\to 0$.\n\n## Remarks\n\nThe [[one-sided derivatives|0HXEX0DPRA]] at $0$ exist: the right derivative is $1$ and the left derivative is $-1$.\nSo the converse of [[differentiable functions are continuous|07ZMX4ANJS]] fails.\nFor $x \\ne 0$, the derivative is $x / |x|$, the sign of $x$. Mathlib states the failure at $0$ as `not_differentiableAt_abs_zero`.\n\n## Explanations\n\n### L1 human\n\nThe graph of $|x|$ is a V with a sharp corner at the origin.\nAt the corner, no single line fits the graph, so there is no tangent.\n",
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