Bijective & continuous -> differentiable?

In summary, a bijective continuous function from [a,b] to [f(a),f(b)] must be differentiable. This is because the intermediate value theorem states that the function must take all values between f(a) and f(b) for any x in [a,b]. Additionally, a function can only be non-differentiable at a set of points of measure zero, which includes vertical tangents. However, there are counter-examples, such as the function f(x) defined on [0, 2], that show that a continuous bijection can be differentiable only on a set of measure zero. This can be further proven using concepts like Lebesgue's decomposition theorem, absolutely continuous functions, and singular measures. The
  • #1
lolgarithms
120
0
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 got to 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)
 
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  • #2
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.
 
  • #4
lolgarithms said:
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 got to 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)
Consider y(x)= x if [itex]0\le x< 1[/itex], 2x if [itex]1\le x\le 2[/itex]. That is a bijective from [0, 2] to [0, 4] and is continuous. It is NOT differentiable at x=1.
 
  • #5
HallsofIvy said:
Consider y(x)= x if [itex]0\le x< 1[/itex], 2x if [itex]1\le x\le 2[/itex]. That is a bijective from [0, 2] to [0, 4] and is continuous. It is NOT differentiable at x=1.
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
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
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

[tex] f(x) = x + c(x) [/tex]​

where [itex] c(x) [/itex] 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
the question mark function -
ouch, i was wrong!
 

1. What is the definition of a bijective function?

A bijective function is a type of function in which each element in the domain is mapped to a unique element in the range, and each element in the range is mapped to by exactly one element in the domain.

2. Can a bijective function be continuous?

Yes, a bijective function can be continuous. A function is considered continuous if there are no abrupt changes or breaks in the graph of the function. A bijective function can be both one-to-one and onto, satisfying the conditions for continuity.

3. What is the relationship between bijectivity and differentiability?

A bijective function is not necessarily differentiable, but a differentiable function is always bijective. A bijective function can be thought of as the combination of two one-to-one functions, one of which is the inverse of the other. In contrast, a differentiable function can be thought of as a function that has a well-defined tangent line at every point on its graph.

4. How can we determine if a function is differentiable?

A function is differentiable if it has a well-defined derivative at every point in its domain. This means that the limit of the difference quotient, or the slope of the tangent line, exists at every point. In order for a function to be differentiable, it must also be continuous.

5. Why is bijectivity important in mathematics?

Bijectivity is important in mathematics because it allows for the concept of one-to-one correspondence between elements of two sets. This is useful in many areas of mathematics, including set theory, combinatorics, and topology. Bijective functions also have useful properties, such as being invertible and preserving the cardinality of sets.

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