Simply put, there is NO such thing in astrology as 'cusp' if you are talking about the sun signs and when they change. If you look up the word cusp in a dictionary it will say Apex, point and the sun signs change at a 'specific point' in time. Not always the same each year either. That is why your 'time' of birth is to very important.

Problem is lots of people don't know there time of birth, if born in UK in Scotland or a twin your time is put on your birth certificate and when computers became commonplace, most hospitals or GP surgeries should have your time of birth.

There is far far more to Astrology that just Sun signs.
Basic astrology
Now basically your sun sign is your 'core beliefs, values, ego' and Asc is your physical body, mode of expression and mask and persona that we all wear and hide behind, but definitely not what's going on underneath. What's going on underneath is your Sun sign. Moon sign, house placements and aspects describe your emotional and intuitive responses, also describes your view of your mother, as saturn describes your father by sign and house position.

So, think of a glass milk bottle the bottle is your Asc and your sun the milk, but the milk still has to 'come out' through the glass neck of the milk bottle....

Now aspects within a natal chart represent 'facets' of our personality, because we are all multifaceted beings. What these do is 'break down and compartmentalise' these facets into smaller bite size chunks. This does take quite a lot of self analysis and objectivity to 'see yourself' and why you do certain things and behave in certain ways. Actually it can be quite good therapy...

I digress, there is a transitional phase going from one aspect to the next. Like it's not bright and sunny all day and then like someone hit a light switch it's star light sky. It's like it gets brighter progressively and whence it is at it's peak, it begins to wane, when the sky is brightest is when it begins to darken. You've become to accustomed to light switches and clocks if you think it just shifts from one to the other with no in between. Return to natural time and your understanding will deepen, clocks and the gregorian calendars are completely arbitrary so following them gives false impressions and creates an illusion of separation from nature causes one to forget how natural time works.

IF you are talking about just sun sign astrology, it's like saying when is a sun sign not a true sun sign.... when it's early or late, cos there is every chance of having a planet ie: venus or mercury in the sign before or after..

With any aspect in astrology it's the position of one cosmic body in relation to another from the perspective of an earth location. For there to be a time when the aspect is exact, like the moon being directly between earth and a star formation there also must be the time prior to and preceding the exact alignment where the point of reference(for a moon sign the moon, for a sun sign the sun) is directly lined up somewhere between one aspect and the another. The time in between is transitional and is the waning of one aspect giving way to waxing of another. When an alignment is exact is when the energy of it is most direct but there is a ripple effect before and after the alignment, and a transitory phase where one aspect recedes well another becomes  more prominent till it peaks and it too begins to recede as another rises.

perhaps you are confusibg astromony with astrology and talking very flowery.This thread is discussing house cusps and sun signs changing at certain points in 'time'

You seem to be discussing 'aspects' which are the mathematical degrees which one planet makes to another or to the Asc/Desc or MC/IC. Yes there are approaching aspects and departing aspects in these by way of assessment