prove that integral

Question

prove that

$$-\int_0^t \ sgn(f({s})) \ d{s}=\int_0^t \ sgn(-f({s}))\ d{s}+2\int_0^t 1_{f({s}) =0}d{s}$$ with $$ sgn(x) := \begin{cases} -1 & \text{if } x =< 0, \\ 1 & \text{if } x > 0. \end{cases}$$

I would appreciate it enormously if anyone could help best,educ

Answer 1

Notice that for all $x$, $\operatorname{sgn}x + \operatorname{sgn}(-x) = -2 \cdot 1_{\{0\}}(x)$.

It follows that:

$$\operatorname{sgn}f(x) + \operatorname{sgn}(-f(x)) = -2 \cdot 1_{\{0\}}(f(x)) = -2 \cdot 1_{f^{-1}\{0\}}(x)$$

Integrating over $[0,t]$ completes the answer.