Let $X$ be the space of continuous and bijective functions $f$, such that $$ f\colon [0,1] \to [0,1] \quad , \quad f(0)=0 \quad , \quad f(1)=1 \, .$$ Is $X$ complete (under the supremum norm $ \|.\|_\infty$ )?
I can show that if instead of bijectivity we only require surjectivity then it is complete, but I have not been able to do so if we also require injectivity ( I'm not sure if it is still complete).
No, a sequence of continuous bijections can converge uniformly to a function that isn't one-to-one. Let $f_n$ be the piecewise linear function obtained by linear interpolation between the points $(0,0), (\frac14,\frac12-\frac1n),(\frac34,\frac12+\frac1n),(1,1)$. For each $n\geq3$, this is a bijection, but the uniform limit is constant on the interval from $\frac14$ to $\frac34$.