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The 11 pages that link to Bounded function, each with the reason it gives.

• Riemann integralRelated: Boundedness keeps the sums finite and is part of the standard definition.

• IntegralRelated: Boundedness helps guarantee Riemann integrability on a closed interval.

• Supremum normRelated: Boundedness makes the supremum norm finite.

• Liouville's theoremRelated: A global bound on the entire function is the theorem's decisive hypothesis.

• Bounded setRelated: The distance from a fixed point turns boundedness of a set into boundedness of a distance function.

• Riemann sumRelated: Boundedness is part of the standard criterion for Riemann integrability.

• Removable singularityRelated: Local boundedness near an isolated singularity guarantees holomorphic extension.

• Maximum and minimumRelated: Boundedness guarantees finite bounds, but does not ensure either bound is attained.

• Weierstrass theoremRelated: Compactness and continuity first ensure the function’s values cannot escape to infinity.

• Tietze extension theoremRelated: The bounded form is the core extension result used to handle arbitrary real-valued functions.