Suppose we have a differentiable function $ F : \mathbb R^n \rightarrow \mathbb R^n$ and it's derivative 'function(linear function)' at each point always has non zero determinent then is the function onto? I know it is one-one and onto in small domain but can we extend it over all to whole domain?
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No. For instance, the function $f: \Bbb R \to \Bbb R$ defined by $$f(x) = e^x$$
has non-zero derivative but is not onto.
General Grievous
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