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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be Riemann integrable.\nDefine $F(x) = \\int_a^x f(t) \\, dt$ for $x \\in [a, b]$.\nThen $F$ is continuous on $[a, b]$.\nIf $f$ is continuous at $c \\in (a, b)$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.\n\n## Remarks\n\nIf $|f| \\le M$, then $|F(y) - F(x)| \\le M |y - x|$. So $F$ is Lipschitz continuous.\nBy [[additivity|05RCDKHZM6]], $\\frac{F(c + h) - F(c)}{h}$ is the average of $f$ over the interval between $c$ and $c + h$. Continuity at $c$ makes this average tend to $f(c)$.\nMathlib states a strict form as `intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right`.\n\n## Explanations\n\n### L1 human\n\nLet the area under the graph from $a$ to $x$ be a function of $x$.\nWhere $f$ is continuous, this area grows at the rate $f(x)$.\n",
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