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'?
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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 $.
Or, use f(x)= 1 for all x. That will "smooth" any function!
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 $\displaystyle x=0$.
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