2 (vii) if lim f(x) = L then lim kf(x) = kL, for any constant k. 4 Continuity For a function,[(x), to be continuous at some point, say at x =a, we require that f(x) should approach the value f(a) as x approaches a from either side. Having dealt with the idea of a limit, we may make the following definition. Definition 2. The functionf(x) is continuous at x =a if f(x)~f(a)asx~a Note that this implies, (i) that lim f(x) exists and is finite, and x--+o (ii) that f(a) is defined and has as its value the value of this limit.
Let [(x) be defined and continuous on a finite, closed, interval a~ x ~ b, and let m and M denote respectively its greater lower bound and its least upper bound on that interval. Cx1) =m and [(x2) (b) =M if c is any number lying between m and M then there is at least one value of x in a