Suppose that f and g are 1-1 on R. If f and g◦f are continuous on R, then g is continuous on R.
The answer is false, according to the solution, but I can't see why..
My initial reasoning was this :
Let h := g◦f. By the hypothesis, h is continuous.
Also, since f is 1-1 and continuous on R, $f^-1$ is defined and continuous on R.
Hence, h◦$f^-1$ is defined on R, and continuous on R (by the composite property of continuity).
Therefore, g = h◦$f^-1$ is continuous on R.
Where does my reasoning go wrong?
Thanks!
It depends on the range of $f$. If Range $f \in$ Domain $g$, then $g◦f$ is continuous. However, Range $f$ is not necessarily the whole Domain of $g$. So for any value $x \in$ Domain $g$ and $x$ not in Range $f$, $g$ could be 1-1 and not continuous.