I'd like some hint for the following question:
Let $ f:I \longrightarrow \mathbb{R} $ monotone on a interval $I$. If the image $f(I)$ is a interval, show that $f$ is continuous.
I'd like some hint for the following question:
Let $ f:I \longrightarrow \mathbb{R} $ monotone on a interval $I$. If the image $f(I)$ is a interval, show that $f$ is continuous.
A very informal answer will follow. You may wish to fill in the details your self. Note that if $f(x)$ is monotone on some interval, then both the left and right hand limits of $f$ exists at every point of the interval. Hence if $f$ is discontinuous at some point then the left and right hand limit at this point cannot agree. But this in turn implies that $f(I)$ is not an interval.
HINT: If $f$ is monotone, the only kind of discontinuity that it can have is a jump discontinuity. (You’ll have to show this, but it’s not too hard.) If that happens, can $f[I]$ be an interval?