# Continuity of composite function

• Nov 26th 2011, 01:55 AM
deepak
Continuity of composite function
Is it possible for a composite function f(g(x)) to be continuous at a point 'x=a'
when g(x) is not continuous at 'x=a'?
• Nov 26th 2011, 03:20 AM
Plato
Re: Continuity of composite function
Quote:

Originally Posted by deepak
Is it possible for a composite function f(g(x)) to be continuous at a point 'x=a' when g(x) is not continuous at 'x=a'?

Let $\displaystyle g(x) = \left\{ {\begin{array}{*{20}c} { - 1,} & {x \leqslant a} \\ {1,} & {x > a} \\ \end{array} } \right.\;\& \,f(x) = x^2$.
• Nov 26th 2011, 12:33 PM
HallsofIvy
Re: Continuity of composite function
Or, use f(x)= 1 for all x. That will "smooth" any function!
• Nov 26th 2011, 12:48 PM
alexmahone
Re: Continuity of composite function
Quote:

Originally Posted by HallsofIvy
Or, use f(x)= 1 for all x. That will "smooth" any function!

If $\displaystyle f(x)=1$ and $\displaystyle g(x)=\frac{1}{x}$, $\displaystyle f(g(x))$ is not continuous at $\displaystyle x=0$.