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Homework Statement
Let a>b be Real numbers and
f, g: [a,b] --> R be continuous and differentiable on (a,b)
Show g is injective on [a,b] if g'(x) != 0 for all x in (a,b)
Homework Equations
Rolle's theorem: Continuity and differentiability (in the conditions above) imply that
f(a) = f(b) and there exists c in (a,b) such that f'(c) = 0
The Attempt at a Solution
Well first I don't know exactly what injective means (what is "distinctness"). What I do understand is Rolle's theorem: that there is a turning point or point of zero gradient between any two points that have the same y-value (if that's right). So in this question there is no turning point or zero gradient in the interval [a,b] - but I don't know what the function is restricted to look like. I'm thinking it could be a horizontal straight line, a parabola, or a squiggly thing that starts and ends between two horizontal points. I'm really quite clueless how to prove something for all situations
If you could just give me a starting point or outline,
Thanks