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  "body": "\n## Proof\n\nWe combine the two parts of the theorem.\n\n1. Let $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be continuous. Since [[continuous functions are Riemann integrable|06SESV7GQY]], $f$ is Riemann integrable on $[a, b]$.\n2. For $a \u003c x \\le b$, $f$ is continuous on $[a, x]$, hence integrable there. So $F(x) = \\int_a^x f(t) \\, dt$ is defined on $[a, b]$, with $F(a) = 0$. Let $c \\in (a, b)$. The function $f$ is continuous at $c$.\n3. By the [[first fundamental theorem of calculus|063H5KJC69]], $F$ is differentiable at $c$ with $F'(c) = f(c)$. So $F$ is differentiable on $(a, b)$ with $F' = f$.\n4. Now let $G$ be continuous on $[a, b]$ with $G' = f$ on $(a, b)$. Then $G$ is differentiable on $(a, b)$, and $f$ is a Riemann integrable function on $[a, b]$ that extends $G'$, by step 1.\n5. By the [[second fundamental theorem of calculus|0XYAFCN0PT]], $\\int_a^b f(x) \\, dx = G(b) - G(a)$.\n\n## Idea\n\nA continuous function is integrable and continuous at every point. So both halves of the theorem apply to it directly: the integral up to $x$ has derivative $f$, and any antiderivative computes the integral.\n",
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