I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe Shilov briefly touches on this in his "Elementary Real and Complex Analysis", but I'd like a more broad and in-depth treatment - book that, in general, deals with mostly functions (and maybe continuity/their limits).
Meant for a freshman mathematics major. Books that are entirely dedicated to this topic are preferred over others wherein this topic is only treated in passing.