In linear response theory, the dielectric response at zero frequency sometimes appears to violate the f-sum rule, which has apparent implications for causality.

Key words: electron gas; f-sum rule; Lindhard dielectric function; linear response theory; second order pole; transverse dielectric function.

is known variously as the f-sum rule, the oscillator-strength sum rule, and the Thomas-Reiche-Kuhn sum rule.

The f-sum rule should apply to all response functions that satisfy Eq.

It is even stated in the literature that the transverse dielectric function [[epsilon].sup.(t)] satisfies the f-sum rule of Eq.

(13), it is possible to form the f-sum rule integral of Eq.

This example turns out to be the only relation that is not widely discussed which is needed to complete the discussion of the f-sum rule for Lindhard's transverse dielectric function.

The tradition point of view is that the f-sum rule fails and the formula must be rewritten to include a pole term explicitly.

The key requirements for the f-sum rule are causality, which implies there are no poles in the upper-half complex frequency plane, and the free-electron response at high frequencies.