To say that Duke Orsino insists on continuing to call Viola "Cesario" at the end of Shakespeare's Twelfth Night is inaccurate and somewhat misleading. Duke Orsino addresses Viola as "Cesario" only once in the entire fifth act.
Once "Cesario" reveals that "he" is actually Viola, Orsino instantly transfers his love for Olivia to Viola, if for no reason other than that he's developed a fondness for "Cesario," who is now Viola, and Viola professes her love for him. Orsino then proposes to Viola.
DUKE ORSINO. Your master quits you; and, for your service done him,
So much against the mettle of your sex,
So far beneath your soft and tender breeding,
And since you call'd me master for so long,
Here is my hand: you shall from this time be
Your master's mistress. (5.1.329–334)
When the mistaken identities have been revealed and all of the merry mix-ups have been sorted out, Orsino addresses Viola once more as "Cesario."
DUKE ORSINO. Cesario, come:
For so you shall be, while you are a man;
But when in other...
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