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  "body": "\n## Proof\n\nWe restrict $f$ to the segment from $x$ to $y$ and apply the one-variable mean value theorem.\n\n1. Let $U$, $f$, $x$ and $y$ be as in the statement. Define $\\gamma(t) = x + t (y - x)$ for $t \\in [0, 1]$. By hypothesis, $\\gamma(t) \\in U$ for each $t \\in [0, 1]$.\n2. Let $z \\in U$. By [[linear maps on finite-dimensional spaces are continuous|0C13E623WD]] and [[bounded linear maps are continuous|0F3K4C7XYW]], there is $C \\ge 0$ with $|Df(z) h| \\le C \\|h\\|$ for each $h$.\n3. Since $f$ is [[differentiable|0135XZZQ8M]] at $z$, there is $\\delta \u003e 0$ with $|f(z + h) - f(z) - Df(z) h| \\le \\|h\\|$ for $\\|h\\| \u003c \\delta$ and $z + h \\in U$. By the [[triangle inequality|0FJZMBX0ZZ]], $|f(z + h) - f(z)| \\le (C + 1) \\|h\\|$ for these $h$. So $f$ is continuous at $z$.\n4. Put $g(t) = f(\\gamma(t))$ for $t \\in [0, 1]$. Since $\\|\\gamma(t) - \\gamma(t')\\| = |t - t'| \\, \\|y - x\\|$, the map $\\gamma$ is continuous. By step 3, $g$ is a [[composition of continuous maps|05NK26G0DR]]. So $g$ is continuous on $[0, 1]$.\n5. For $t \\in (0, 1)$ and small $s \\ne 0$, $(\\gamma(t + s) - \\gamma(t)) / s = y - x$. So $\\gamma$ is differentiable on $(0, 1)$ with $\\gamma'(t) = y - x$.\n6. By the [[chain rule for a composition with a curve|0A8TJ0VHC7]] on the open interval $(0, 1)$, $g$ is differentiable at each $t \\in (0, 1)$, with $g'(t) = Df(\\gamma(t))(y - x)$.\n7. By the definition of the [[gradient|0NS2QYVC4Y]], $g'(t) = \\nabla f(\\gamma(t)) \\cdot (y - x)$ for $t \\in (0, 1)$.\n8. By the [[mean value theorem|0BARQY52N5]] for $g$ on $[0, 1]$, there is $c \\in (0, 1)$ with $g(1) - g(0) = g'(c)$.\n9. Since $g(1) = f(y)$ and $g(0) = f(x)$, steps 7 and 8 give $f(y) - f(x) = \\nabla f\\big(x + c (y - x)\\big) \\cdot (y - x)$.\n\n## Idea\n\nAlong the straight segment from $x$ to $y$, the function $f$ becomes a function $g$ of one variable $t$. The one-variable mean value theorem gives a point where the slope of $g$ is its average slope. The chain rule identifies that slope with the gradient dotted with the direction $y - x$.\n",
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