Understanding Continuity in Functions: Quick FAQs

[jason177]

Lets say you have a function f(x)=1/x-1/x+x this function would still be discontinuous at x=0 even though the 1/x's would cancel, right? Also I know that combinations of continuous functions are also continuous, so for example if f and g are continuous then f+g is continuous. So my other question is: does that go both ways? so if you know that f=g+h is continuous does that imply that g and h are both continuous?

 

[wisvuze]

it would be discontinuous at 0, since 1/x is not even defined at zero, so it wouldn't "make sense" to write f(0 ) if you had a 1/x term involved

if f = g+ h is continuous it certainly doesn't imply that g and h are continuous as well. For example, try g(x ) = 1 if x is irrational and g(x ) = 0 if x is rational -- then set h to be vice versa, g + h will be a constant function ( with the constant 1 ) but it is clear that neither g and h are continuous everywhere