f(x)Il.PropertiesofFunctionsRD.1. BoundednessDef: We say f(x) is bounded above (below)onXifVx E X,3M > 0, suchthat f(x)≤ M(f(x)≥-M)We say f(x) is bounded on XifIf(x)/≤ M.Vx E X,3M > 0, such thatMxMS1.2 Functions
§1.2 Functions II. Properties of Functions 1. Boundedness D R ⎯f ⎯( x) → Def: x X,M 0, such that f (x) M( f (x) −M). We say f (x) is bounded above (below) on X if x X,M 0, such that f (x) M. We say f (x) is bounded on X if -M y x o M N
f(x)II. Properties of FunctionsRD2. MonotonicityDef: X c D, Vxi,x, E X, Xi <x2,thenfis increasingonX if x, <x, = f(x)< f(x2)fis decreasingonX if xi<x,=f(x)>f(x,)10n (0,+)Forexample,y=x0n (-00,0) U (0,+0)S1.2 Functions
§1.2 Functions 2. Monotonicity then f is increasing on X if ( ) ( ) 1 2 1 2 x x f x f x f is decreasing on X if ( ) ( ) 1 2 1 2 x x f x f x Def: , , , , X D x1 x2 X x1 x2 For example, on (−,0)(0,+) on (0, ) 1 = + x y II. Properties of Functions D R ⎯f ⎯( x) →