I am not really a fan of film Denethor (although I do think that John Noble plays the character as written superbly). But, realisitically speaking, there was not enough time within the last film to establish Denethor as the proud but noble character we meet initially in the book and then have him change into the suicidal and infanticidal character that he is at his death. The contrast would not have come across credibly within the time available to devote to him on screen. And so it made sense to have him as an unsympathetic character from the outset.
But I think that there is a change in his character, albeit subtle, throughout the film. As
Boromir88 points out, he only becomes really psycho once he sees the size of the force approaching Minas Tirith and then is given to believe that his remaining son is dead.
As for the original question, I think that there is sufficient in the film to establish the credibility of his despair, even without the Palantir. He is fully aware that Minas Tirith cannot hope to hold out on its own against Sauron without having to be told as much by Sauron himself. Add to that the death of his favoured son and his conduct with regard to the defence of Minas Tirith and his attitude towards Gandalf (supplemented by his credible reluctance to give up his throne to a "usurper") seems believable to me. The reality of Sauron's assault on the White City and the death of his remaining son (or so he believes) is credibly sufficient to push him over the edge (if you'll excuse the pun
).