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Calculus Continuity question?
If f is continuous and 0 <= f(x) <= 1 for x belongs to [0, 1], then prove that there exists c in [0, 1] such that f(c) = c.
1 Answer
- Awms ALv 71 decade agoFavorite Answer
If f(0) = 0 or f(1) = 1, then we have our c already, so we might as well assume that f(0) > 0 and f(1) < 1.
Now consider g(x) = f(x) - x.
g is continuous, g(0) = f(0) - 0 > 0, and g(1) = f(1) - 1 < 0.
Thus by the intermediate value theorem, there's a c in (0,1) such that g(c) = 0.
In other words, f(c) - c = 0, from which we get
f(c) = c.
