1549901369-Elements_of_Real_Analysis__Denlinger_

626 Appendix B • Sets and Functions

We define algebraic operations on this set :F(S, JR). In particular, we define

(a) Addition: Vf,g E :F(S, JR), we define the function f + g by

specifying that Vx E S ,

(f + g)(x) = j(x) + g(x).

"-.,---" "-,-"'

in :F(S, JR) in JR

(b) Multiplication by "scalars" r E JR: VJ E :F(S, JR), and

Vr E JR, we define the function r f by specifying that Vx E S,

(rf)(x) = r · f(x).

._.,......, "-.,---"

in :F(S, JR) in JR

(c) Multiplication: Vf,g E :F(S, JR), we define the function Jg

by specifying that Vx E S ,

(fg)(x) = f(x) · g(x).

._.,......, ~

in :F(S, JR) in JR

(d) Division: Vf,g E :F(S, JR), we define the function !_ by

g

specifying that Vx E S ,

(~) (x )

'-v-'

in :F(S, JR)

= (f(x))

g(x)

'--,.--"

in JR

Notice that f jg is not necessarily in :F(S, JR), since we do not

know if the denominator is ever 0 without knowing the specific

function g(x). The domain off /g may be different from S.

(e) Absolute value: VJ E :F(S, JR), we define the function lfl

by specifying that Vx E S,

lfl (x) = lf(x)I.

._.,......, '-v-'

in :F(S, JR) in JR

(f) Maximum: VJ, g E :F(S, JR), we define the function max{!, g}

by specifying that Vx E S ,

max{f, g}(x ) = max{f(x),g(x)}.

'--,,.--'

in :F(S,JR) in JR