Does there exist a continuous injective or surjective function from $[0,1)$ to $(-1,1)$ ? I know there is no continuous bijection from $[0,1)$ to $(-1,1)$ , but am stuck with only injective continuous or surjective continuous . Please help . Thanks in advance
I think next two functions are continuous and:
$f(x) = x$ is injective but not surjective
$g(x) = x * \sin(1/(1-x))$ is surjective but not injective
$$ f(x) = x. $$ This is a continuous injection from $[0,1)$ onto $[0,1)$ and so a continuous injection from $[0,1)$ into $(-1,1)$.
For a continuous surjection, let's do it piecewise. Say a function $g$ is piecewise linear and $g(0)=0$, $g(1/2) = 0.9$, $g(3/4)=-0.9$, $g(7/8) = 0.99$, $g(15/16) = -0.99$, $g(31/32) = 0.999$, $g(63/64)=-0.999$, and so on. The image of this continuous function is all of $(-1,1)$.