Thread: Composite functions continuity theorem

g: R->R

g(x) = 2 if x not equals 1
g(x) = 0 if x equals 1

f(x) = x +1 for R

"Verifiy" (sorry my english) that lim x-> 0 (gof)(x) is not equal lim x->0 (gof)(0)

If there is a contradiction with the continuity theorem of composite function, give a justification why.



------------------------------------------------------------------

Firs part:

lim x->0 (gof)(x) = lim x->0 g(x+1) = g(1) = 2
lim x->0(gof)(0) = g(0+1) lim x->0 g(1) = 0

??

and now what is continuity theorem of composite function?

Originally Posted by Fabio010

g: R->R
g(x) = 2 if x not equals 1
g(x) = 0 if x equals 1
f(x) = x +1 for R
"Verifiy" (sorry my english) that lim x-> 0 (gof)(x) is not equal lim x->0 (gof)(0)
If there is a contradiction with the continuity theorem of composite function, give a justification why.
lim x->0 (gof)(x) = lim x->0 g(x+1) = g(1) = 2
lim x->0(gof)(0) = g(0+1) lim x->0 g(1) = 0

The parts in red are not the way it is usually done.

BUT shows that is not continuous at

Hum so gof is not continuous in x=0.

But that contradicts the theorem of continuity?

The only thing i know is that:

if ( is continuous in and is continuous in then if continuouse in )

Originally Posted by Fabio010

Hum so gof is not continuous in x=0.
But that contradicts the theorem of continuity?

But in this case is not continuous at .
but .

Originally Posted by Fabio010

if ( is continuous in and is continuous in then if continuouse in )

That should be is continuous at .