Relation with Classical Derivatives

Let’s see how *f*′ is related to *Df* when *f* is a piecewise smooth function whose classical derivative, *f*′, is exponentially integrable. For now, assume *f* has exactly one discontinuity, say, a jump of *j*_{0} at *x*_{0}. Remember,

${j}_{0}=\underset{x\to {x}_{0}^{+}}{\mathrm{lim}}f\left(x\right)-\underset{x\to {x}_{0}^{-}}{\mathrm{lim}}f\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$

At this point we should observe that *f*(*x*) can be computed from its derivative via

$f\left(x\right)-\underset{s\to {x}_{0}^{+}}{\mathrm{lim}}f\left(s\right)={\displaystyle {\int}_{{x}_{0}}^{x}f\left(s\right)ds}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{0}<x$

and

$\underset{s\to {x}_{0}^{-}}{\mathrm{lim}}f\left(s\right)-f\left(x\right)={\displaystyle {\int}_{x}^{{x}_{0}}f\prime \left(s\right)ds}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<{x}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$

From this and the exponential integrability of *f*′ we can easily deduce that *f*, itself, must be exponentially bounded on the real line (see lemma 30.10 on page 496).

Now let *ϕ* be any Gaussian test function. By definition,

$\langle D\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi \rangle =-\langle f,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi \prime \rangle =-{\displaystyle {\int}_{-\infty}^{\infty}f\left(x\right)\varphi \prime \left(x\right)dx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$ |
(35.26) ... |

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