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Jun 21 at 16:08 comment added Akira Thank you very much for your detailed answer. The example of $f$ is very illustrative in showing that $h \downarrow 0$ could not make $H(z) \downarrow 0$. This problem is due to the discontinuity of $f$. The smaller $\delta$, the more severe the discontinuity of $f$.
Jun 20 at 15:17 vote accept Akira
Jun 20 at 15:08 history edited Iosif Pinelis CC BY-SA 4.0
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Jun 20 at 15:01 history edited Iosif Pinelis CC BY-SA 4.0
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Jun 20 at 14:51 history answered Iosif Pinelis CC BY-SA 4.0