Every computational task can be solved with $O(1)$ additional memory

Question

We were wondering, is it true that no matter which complexity we solve the problem with, it can be solved using $O(1)$ additional memory? Does it refer to a kind of unsolved computational math problem or can be easily proofed (disproofed)?

Answer 1

The space hierarchy theorem proves that this is false. In fact, it shows that for a large class of functions $f$, there are problems which can be solved using $O(f(n))$ bits of memory, but not $o(f(n))$ bits.

Just take $f(n)$ to be any function that grows with $n$.