Darboux's Theorem/Corollary
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Corollary to Darboux's Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Suppose that $\forall t \in \closedint a b: \size {\map f t} < \kappa$.
Then:
- $\ds \forall \xi, x \in \closedint a b: \size {\int_x^\xi \map f t \rd t} < \kappa \size {x - \xi}$
Proof
Follows directly from Darboux's Theorem.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.6$