Question

th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any e >0, there exists an open set U of E such that aEU and sup f(x)-inf,f(x) < ε

Answer 1

on- iven that 5) ion deftoed DCR, to be Contiusor A Tunction cle- 1. then 0 let Ir-014 S ohere S is oise ket "d Satisfies 1,-011S and the valus of ;s then let e3 5) 2) is Continuou at x c than 3 S>o , Such that

now Ca)-V inc e orbita-for REU the elation holds truu ain let CConverse SU inffa) LE ョ an open ps C) Set E Such that -the relation ad tius

e Cat the abov relatior xe u Ev fex) lenuu F is continuous at χ:a.