As we know, if $f:R→R$ is continuous, and $F(x)=\int_0^xf(y)dy$, then $F$ is differentiable and $F'(x)=f(x)$. (Fundamental Theorem of Calculus)

Consider the case that $f$ is not continuous, let $F(x)=\int_0^xf(y)dy$. Assume $f$ is increasing thus $F$ is convex. Now $F$ doesn't have to be differentiable but it has subgradients (subderivatives). Is there any relationship between $f$ and subgradient of $F$?

Is it true? Can we prove this: $c$ is a subgradient of $F(x)$, if for any $y<x,f(y)<c$ and for any $y>x, f(y)>c$.

Here is an example to justify this proposition. Suppose $f(x)=1$ for $x>=0$, $f(x)=-1$ for $x<0$. $F(x)=\int_0^xf(y)dy=|x|$. The proposition holds in this example.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.