Preface

Our live s an d th e univers e barel y work , bu t that' s OK ; it' s amazin g an d

great tha t the y wor k a t all . I thin k i t ha s somethin g t o d o wit h math ,

and especiall y rea l analysis , the theor y behin d calculus , which barely works .

Did yo u kno w tha t ther e ar e function s tha t ar e no t th e integra l o f thei r

derivatives, an d tha t a functio n wit h a positiv e derivativ e ca n decrease ?

But i f you're a littl e carefu l yo u ca n ge t calculu s t o work . You'l l see .

The theor y i s hard , subtle . Afte r Newto n an d Leibni z invente d th e

calculus in the late 1600s, it took puzzled mathematicians two hundred years,

until th e latte r 1800s, t o ge t th e theor y straight . Th e powerfu l moder n

approach usin g ope n an d close d set s cam e onl y i n th e 1900s. Lik e man y

others, I foun d rea l analysi s th e hardes t o f the mat h majo r requirements ; i t

took m e hal f th e semeste r t o catc h on . S o don' t worry : jus t kee p a t it , b e

patient, an d hav e fun .

This text i s designed fo r students . I t present s the theoretical intellectua l

breakthroughs whic h mad e calculu s rigorous , bu t alway s wit h th e studen t

in mind . I f a shortcu t o r som e mor e advance d comment s withou t proo f

provide bette r illumination , w e take th e shortcu t an d mak e th e comments .

The resul t i s a complet e cours e o n rea l analysi s that fits comfortabl y i n on e

semester.

vn