Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 4 / c / Solution

Part (b) shows that $s\mapsto {\tau }_s$ is finite and continuous. Since both $t\mapsto B_t$ and the power function $x\mapsto |x{|}^{\delta }$ are continuous, their compositionis continuous. This is an instance of a time change of a continuous process.