Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function?
I thought about this problem today and I've no idea how to solve it or even attack it.
Also, I would be interested in variants of this problem, in which we substitue $\phi$ for another number-theoretical interesting functions, like $\sigma,\tau,\lambda$, etc.
Thank you in advance!