Determine if the following problem is decidable or not : Does the read–write head of a TM with the input w leave the word w?

Determine if the following problem is decidable or not : Does the read–write head of a TM with the given input w leave the word w on the tape?

It is not clear if your language $$L$$ contains the pars $$\langle T, w\rangle$$ or if $$w$$ is fixed and $$L$$ only contains Turing machines. In any case the answer is the same.
Let $$T$$ be a (single tape) TM that does not leave the word $$w$$. Let $$Q$$ and $$\Gamma$$, be the set of states and the tape alphabet of $$T$$.
The number of possible configurations of $$T$$ can be upper bounded as a function of $$|w|$$, $$|Q|$$, and $$|\Gamma|$$ and this upper bound $$B_T$$ is computable.
To decide $$L$$ it suffices to simulate $$T$$ for $$B_T$$ steps. If $$T$$ ever leaves the word $$w$$ you can reject. Otherwise either $$T$$ halted without leaving $$w$$, or $$T$$ is stuck in an infinite loop that does not leave $$w$$. In both cases you can accept.
• |w| is not an upper bound. I just said that there is an upper bound which can be expressed as a function of $3$ parameters: $|w|$, $|Q|$, and $|\Gamma|$ and is computable. Jun 27 '20 at 20:47