Darboux's Theorem/Corollary

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): $\S 13.6$