how did they know that the trigonmetric function is always continuous at every number in their domain as the polynomials ?

Question

tkhunny · Answer

Why is this "they"?
How do we KNOW polynomials are continuous?

It can be demonstrated, that if you don't wander away too far in one direction (horizontal, maybe), the function will promise not to wander off too far in the perpendicular direction (vertical).  If we carry out this argument to very, very small numbers (arbitrarily small, in fact), this is the definition of continuity.  It is fascinating stuff you should meet in your first calculus course.  It is worth the wait!

anonymous · Answer

what do you mean by horizontal and vertical direction ?

tkhunny · Answer

It is a reference to the graph of a function on a set of coordinate axes.  It is probably the simplest way to think about it at first.  Horizontal means in the direction of changing x values.  Vertical means in the direction of changing y values.