For a monopoly that would find it most profitable to shut down, can price discrimination make it desirable for the monopoly to remain in business?

Yes.

Price discrimination involves consumers being charged differently for the same products. Price discrimination can make it desirable for the monopoly to remain in business due to the following reasons.

1. Through price discrimination, non-profitable firms can't be bankrupt. In some situations, there is no single price that can enable a firm to make profits. Therefore, through price discrimination, some losses are turned into small profits, which allows a firm not to shut down.
2. Investment. When firm price discriminates, they end up getting more profits, which enables them to increase their capacity.
3. Price discrimination supports the principle of equality where low-income earners are expected to pay less, while the rich are charged high prices since they can pay. Low-income earners can benefit from buying products at a lower price.