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Let "support" mean the closed support.

Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function? That is, is there any $f=f(x)$ in one variable with $\operatorname{supp}(f)$ completely contained in $\operatorname{supp}(f')$? Can we choose $f$ smooth?

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1 Answer 1

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It is the contrary. The derivative "decreases" the support. If $x\notin{\rm supp}f$, then for some neighborhood $N_x$ of $x$, $f\equiv0$ in $N_x\implies f'\equiv0$ in $N_x\implies N_x\subset({\rm supp}(f'))^c$. Hence by taking the union over $x\notin {\rm supp}(f)$, $${\rm supp}(f)^c\subset{\rm supp}(f')^c\implies{\rm supp}(f')\subset{\rm supp}(f)$$

A remark: taking the derivative decreases the support strictly if $f$ is nonzero constant in some open neighborhood.

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