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If $f$ is a real valued injective differentiable function from $(a,b)$, then $f’\neq 0$ for all $x\in (a,b).$

I know that $f’(x)=0$ does not imply local constant. Say for example $x^2$. But I am wondering if the above statement is correct?

MathforFun
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2 Answers2

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This is false: consider the injectivity of $f(x) = x^3$ on the real number line.

For any two real numbers $a, b$ we have that $a^3 = b^3$ implies $a^3 - b^3 = 0,$ which we can rewrite as $(a-b)(a^2 + ab + b^2) = 0.$ So, either $a = b$ or $a^2 + ab + b^2 = 0.$ The second term is a quadratic in $a$ with a discriminant of $b^2 - 4(1)(b^2) = -3b^2$ which is negative everywhere except $b = 0,$ where there is one double root at $a = b = 0.$ So, either $a = b$ or $a = b = 0,$ so $a = b$ and $x^3$ is injective. However, the derivative of $x^3$ is $0$ at $x = 0.$

I think the proper statement is that if a differentiable function $f$ is injective on $(a,b)$, then we must have $f' \neq 0$ almost everywhere on $(a,b).$ Otherwise, there would exist some interval $(c_1, c_2)$ such that $\int_{c_1}^{c_2} f'(t) dt = f(c_2) - f(c_1) = 0.$

We can also say that if we have $f' \ne 0$ on a given interval $(a,b)$ then the function is injective over that interval. To show this, assume that there was such a function $f$ defined on $(a,b)$ with $f' \neq 0$ such that there were some $x$ and $y$ in the interval such that $x \neq y$ but $f(x) = f(y).$ By Rolle's theorem (special case of the Mean Value Theorem) we must have that there is some $c \in [x, y]$ such that $f'(c) = 0.$ But by definition, because $[x,y] \subset (a,b), f'(c) = 0$ contradicts our assumption that $f' \neq 0$ everywhere on $(a, b).$

Stephen Donovan
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    $$$$A continuous injection $f:(a,b)\to\Bbb R$ is strictly monotonic, so one can apply the following theorem: $$$$ Let $f:I\to\Bbb R$ be a function differentiable on an open interval $I\subseteq\Bbb R$ and $S={x\in I\mid f'(x)=0}.$ The function $f$ is strictly increasing(decreasing) iff $S$ doesn't contain an open interval and $f'(x)>0$ $(f'(x)<0),\forall x\in I\setminus S.$ $$$$ – PinkyWay Dec 25 '21 at 09:49
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As suggested in the comments just take $f(x)= x^3$ over the real line. Note that it is injective; since it is a polynomial it is even differentiable infinitely many times. But, as you may check $f'(0)= 0$.

David C. Ullrich
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Son Gohan
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