This is a bijection (every element in $(0, 1)$ stays put, and every element in $$ gets sent to $$ by subtracting $1$), so there is an inverse. Other niceties include "linear" for vector spaces, "differentiable" for calculus / analysis and "homomorphism" in abstract algebra.Įxample: you have the function that takes the two intervals $(0, 1)$ and $$ and "glues" them together at one end to make $(0, 2]$. There are bijective, continuous maps where the inverse is not continuous. For instance, in topology, we're mostly interested in continuous maps, so "nice" in that setting means "continuous". It means that not only is there an inverse, but the inverse is "nice" (and the function itself must be "nice" too). The inverse is so-called two-sided, which means that not only can you go there and back again, but you could also start at the other end, go back and then there again. This means that there is an inverse, in the widest sense of the word (there is a function that "takes you back"). Injective and one-to-one mean the same thing.īijective means both injective and surjective.
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