I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked
Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except at time $0.5$ where it jumps by $x$. What are the first and second variations of $\log S_t$ over the period $[0, 1]$.
The first variation is easily determined to be $\infty$, as in a continuous Brownian motion.
A continuous Brownian motion should also have its second variation equal to $T$, so here - not considering the jump - it would be equal to $T = 1$. But unlike the first variation, the second one is a finite value, so should be susceptible to the presence of the jump.
Indeed the solution reported in the book states the second variation to be equal to $1.25$.
Where does this result comes from?