A continuous function having an inverse <=> conditions on a derivative?

In summary, when determining if a function has an inverse using the horizontal line test, it must pass for the function to have an inverse. This implies that the function must not have any extremum, which can be proven by showing that the derivative of a bijective function must be either zero or always positive/negative. However, a strict inequality is necessary for the converse to be true.
  • #1
hb1547
35
0
Sorry for the poorly-worded title.

I help tutor kids with pre-calculus, and they're working inverse functions now. They use the "horizontal line test" to see if a function will have an inverse or not by seeing visually if it's one-to-one.

I was thinking about what that might imply. If a function passes the horizontal line test (let's assume it's domain is ℝ and that it's continuous everywhere), then it must not have any extremum, since then it'd fail the test. So this would imply that:
[tex]f'(x) \le 0 \hspace{3pt}\mathrm{or }\hspace{3pt} f'(x) \ge 0 \hspace{3pt} \forall x[/tex]
I'm wondering if this is always true, and so if it's possible to prove that a bijective function must have a derivative that's either zero or positive/negative everywhere, and I'm wondering if the converse is also true.

I'm more of a physics-guy than a math-guy, but I do find math interesting and these are the types of questions that I like to think about. Any thoughts?
 
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  • #2
I'm wondering if this is always true, and so if it's possible to prove that a bijective function must have a derivative that's either zero or positive/negative everywhere
I would imagine that it is always true for the kinds of functions and inverses you are considering and proving the horizontal line test would be sufficient proof of this too. I think the converse logically follows by completeness - to have an inverse, the derivative of the bilinear form must be definite.
 
  • #3
Simon Bridge said:
I would imagine that it is always true for the kinds of functions and inverses you are considering and proving the horizontal line test would be sufficient proof of this too. I think the converse logically follows by completeness - to have an inverse, the derivative of the bilinear form must be definite.

Consider the function
[itex]f(x)=\begin{cases}(x+2)^3,&x<-2,\\0,&-2\leq x\leq 2,\\(x-2)^3,&x>2.\end{cases}[/itex]
You can see that [itex]f(x)[/itex] is continuous on [itex]\mathbb{R}[/itex] and that [itex]f'(x)\geq0\forall x\in\mathbb{R}.[/itex] However, this function fails the horizontal line test on [itex][-2,2][/itex] and therefore does not have a continuous inverse. A sufficient condition for the converse to be true would be a strict inequality.
 

1. What is a continuous function?

A continuous function is one that has no sudden jumps or breaks in its graph. This means that the values of the function change smoothly and continuously as the input values change.

2. What is an inverse function?

An inverse function is a function that "undoes" or reverses the effect of another function. In other words, if a function f(x) transforms an input x into an output y, the inverse function f^-1(y) will transform the output y back into the original input x.

3. What does it mean for a continuous function to have an inverse?

If a continuous function has an inverse, it means that every output of the function corresponds to a unique input, and every input corresponds to a unique output. This allows us to "undo" the function and get back to the original input value.

4. What are the conditions for a continuous function to have an inverse?

The main condition for a continuous function to have an inverse is that it must be one-to-one, meaning that each input value corresponds to a unique output value. In addition, the function must also be "invertible", meaning that the inverse function must exist and be well-defined.

5. Is the derivative of a continuous function always required for it to have an inverse?

No, the derivative of a continuous function is not always required for it to have an inverse. As long as the function satisfies the conditions mentioned above, it can have an inverse even without having a derivative. However, the derivative can provide useful information about the behavior of the function and its inverse.

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