This is a cross-post from signal processing forum as it was not conclusive. Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then $$ X(\omega) = \sum_{n = 0}^{N-1} x[n] e^{-j \omega n}. $$

The length-$N$ DFT of $x[n]$ is $$ X[k] = \sum_{n = 0}^{N-1} x[n] e^{-j 2 \pi \frac{k n}{N}}. $$ For this, the DFT is a sampled version of the DTFT, i.e., $$ X[k] = X(2\pi k / N). $$ Please also see posts here and here. Now, we consider the maximum magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.

*How much does $m_{\textrm{d}}$ underestimate $m_{\textrm{c}}$? Is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$?*