How does the author reveal Hamlet’s fragile state, despite slips from reality?

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Your question concerning Shakespeare's Hamlet is a bit confusing.  You use the word "despite" even though breaks from reality would definitely reveal a "fragile state."  I don't know that Hamlet suffers any breaks from reality, but since you don't want those anyway, I'll deal with other revelations. 

Hamlet's fragile state is revealed mostly during his speeches.  In Act I he classifies the world as an "unweeded garden."  Hamlet is taking his own situation (unexpected loss of a father, hasty remarriage of his mother, and loss of the crown that he must have considered his--these are revelations of his shaky state, too, by the way) and applying it to the state of the world and of human existence. 

Hamlet also wishes at one point that God had not declared suicide a sin.  Obviously, he is contemplating it.

In the famous "To be or not to be" speech, Hamlet questions the value of existence. 

Hamlet sees all women as unfaithful, lustful, frail, ungrateful, once again applying his own situation (his problems with his mother and Ophelia) to the world as a whole.   

Hamlet also uses sarcasm and satire in his conversations with others (Claudius, Polonius, Gertrude, Ros. and Guil.), possibly a sign of depression.

All in all, Hamlet is depressed and despairing.  He definitely is in a fragile state.

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