It’s a good exercise to think about why the extreme value theorem fails if the function is not continuous (e.g. a general function is more like an infinitely large, arbitrarily malicious lookup table, not something you draw with a pencil on graph paper).
I liked complex analysis better, but I agree that real analysis is generally the first serious math class people who are not very algebraic cut their teeth on.
I think that complex analysis is much more intrinsically interesting (leading to Riemann surfaces, quasi conformal makings, etc,) but because the class of functions is so restricted, one is saved a lot of the nitty-gritty type work that one has to do in real analysis.
It’s a good exercise to think about why the extreme value theorem fails if the function is not continuous (e.g. a general function is more like an infinitely large, arbitrarily malicious lookup table, not something you draw with a pencil on graph paper).
I liked complex analysis better, but I agree that real analysis is generally the first serious math class people who are not very algebraic cut their teeth on.
I think that complex analysis is much more intrinsically interesting (leading to Riemann surfaces, quasi conformal makings, etc,) but because the class of functions is so restricted, one is saved a lot of the nitty-gritty type work that one has to do in real analysis.