How can I find Sobolev generalized derivative of function $f(t)$

Question

Given following function $$ f(t) := \begin{cases} 0 & \text{ if } t \in (-\infty ,1] \\ 2x -2& \text{ if } t \in (1, 2] \\ 2& \text{ if } t \in (2, 3) \\ 8 - 2x& \text{ if } t \in (3, 4] \\ 0 & \text{ if } t \in (4, \infty) \end{cases} $$ How can I find Sobolev generalized derivative of function $f(t)$

Answer 1

In general, if $f$ is continuous and piecewise continuously differentiable (say on the intervals $(-\infty,t_1],[t_1,t_2],\dots,[t_{n-1},t_n],[t_n,\infty)$), then $$ g(t)=\begin{cases}f'(t)&\text{if }t\neq t_k\\0&\text{otherwise}\end{cases} $$ is a weak derivative of $f$. This can be easily seen by splitting up the integral in the definition of weak derivatives and integrating by parts in each of the pieces. I leave it up to you to apply this result to the function at hand.