Proof these exercises.

Exercise 2.LetAR, letf:AR, and letx0A. Thenfis continuous atx0iff given any monotonic sequence (xn) inAwithxnx0, we havef(xn)f(x0).

Exercise3.LetA,BR,letf:AB,g:BRbefunctions,andletx0A. Iffis continuous atx0andgis continuous atf(x0), thengfis continuous atx0.

Exercise 4.Letf:RRbe a continuous function, and letx0R. Iff(x0)>0, then there exists >0 such thatf(x)>0 for allx(x0, x0+).

Exercise 5.Prove the density of the irrationals: givena, bRwitha < b, there existsxR\Qwitha < x < b.