# Bijective & continuous -> differentiable?

1. Aug 3, 2009

### lolgarithms

Is a bijective continuous function:[a,b]->[f(a),f(b)] differentiable?
I think it has to be.
continuity between two distinct values of f(a) and f(b): it gotta take all the values between f(a) and f(b) at x in [a,b], by the intermediate value theorem.
if f is bijective, at [a,b], f(x) can't go up and then down at [a,b]. it has to be monotonically increasing for it to be bijective.
so a function can only be non-differentiable at a set of points of measure zero. (like the vertical tangent of f(x)=x^1/3 at 0)

2. Aug 3, 2009

### CompuChip

The function f defined by:
f(x) = x when 0 <= x <= 1
f(x) = 2x - 1 when 1 <= x <= 2
on [0, 2] is a counter-example to the original statement "bijective & continuous => differentiable".

I suspect you can use this even to make a continuous bijection that is differentiable only on a set of measure zero, by taking a monotonously increasing function like above, of which the slope increases at every non-rational number.

3. Aug 3, 2009

4. Aug 4, 2009

### HallsofIvy

Staff Emeritus
Consider y(x)= x if $0\le x< 1$, 2x if $1\le x\le 2$. That is a bijective from [0, 2] to [0, 4] and is continuous. It is NOT differentiable at x=1.

5. Aug 4, 2009

### CompuChip

It's not continuous, it has a jump at x = 1. If you shift it down by 1, like I did in my example, it works out.

6. Aug 5, 2009

### lolgarithms

i mean... differentiable at all but a "small" (smaller than a real interval) set of points
like a finite set - like the one compuchip mentioned

7. Aug 6, 2009

### DrGreg

First, I can think of a continuous bijection that fails to be differentiable on an uncountable set of measure zero, viz. the Cantor set. It is

$$f(x) = x + c(x)$$​

where $c(x)$ is the Cantor function.

Second, that Wikipedia article links to Minkowski's question mark function, which is claimed to have a zero derivative on the rationals but is not differentiable on the irrationals.

8. Aug 6, 2009

### lolgarithms

the question mark function -
ouch, i was wrong!