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  "body": "\n## Definition\n\nLet $M$ be a $C^1$ [[Banach manifold|9K15AMGBV9]], for example a finite-dimensional smooth manifold.\nLet $f \\colon M \\to \\R$ be a $C^1$ function, with differential $df_p$ at $p \\in M$ in the sense of the [[Fréchet derivative|06EH97V2A7]].\n\n1. A [[critical point|0YTCMFB4HC]] of $f$ is a point $p \\in M$ with $df_p = 0$.\n2. A *critical value* of $f$ is a number $c = f(p)$ with $p$ a critical point; other real numbers are *regular values*.\n3. For $c \\in \\R$ the *sublevel set* is $M^c = \\set{p \\in M : f(p) \\le c}$.\n\n*Critical point theory* studies the critical points and critical values of such $f$ through the topology of $M$ and of the sublevel sets $M^c$.\nA central topic is the change of $M^c$ as $c$ passes a critical value.\n\n## Remarks\n\nThe basic mechanism is a deformation along a gradient flow.\nIf $[a, b]$ contains no critical value and a compactness condition holds, it shows that $M^a$ is a deformation retract of $M^b$.\nSo the topology of the sublevel sets changes only at critical values.\n\nExample: let $M = S^1 \\subset \\R^2$ and $f(x, y) = y$.\nThe critical points are $(0, 1)$ and $(0, -1)$, with critical values $1$ and $-1$.\nThe sublevel set $M^c$ is empty for $c \u003c -1$, a point for $c = -1$, an arc for $-1 \u003c c \u003c 1$, and the whole circle for $c \\ge 1$.\nSo its topology changes exactly at the two critical values.\n\nNon-example: for $f(x) = e^x$ on $\\R$, every real number is a regular value and $f$ has no critical point.\nThe sublevel set $M^c$ is empty for $c \\le 0$ and is the half-line $(-\\infty, \\log c]$ for $c \u003e 0$.\nSo its topology changes at $c = 0$, although $0$ is not a critical value, and $f$ has no minimum.\nThis is possible because the compactness condition, the [[Palais–Smale condition|4724KRNHAT]], fails.\nAlong $x_n = -n$ the values $f(x_n)$ are bounded and $f'(x_n) \\to 0$, but $(x_n)$ has no convergent subsequence.\n\nThe two classical frameworks are Morse theory, through [[Morse functions|KPQ96640ZJ]], and [[Lusternik–Schnirelmann theory|QARH15D2PN]].\n",
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