Why a Continuous Function Cannot Satisfy f(f(x)) -x for All x

In this article, we will use the properties of continuous functions and the behavior of the function to demonstrate why there cannot exist a continuous function f such that f(f(x)) -x for all x.

Step 1: Analyze the Equation

Given the equation:

f(f(x)) -x

We can apply f to both sides of this equation:

f(f(f(x))) f(-x)

Step 2: Substitute and Simplify

Now, using the original equation again, we can express f(f(f(x))):

f(f(f(x))) f(-x)

Thus, we have two expressions for f(f(f(x))):

This gives us the relationship:

f(-x) f(f(x))

Step 3: Check the Implications of Continuity

Assuming f is continuous, we can analyze the implications of the above relationship:

Behavior at Zero

Consider x 0:

f(f(0)) -0 0

This means f(f(0)) 0. Let c f(0), then we have:

f(c) 0.

Behavior at c

Now substitute c back into the original equation:

f(f(c)) -c

Since f(c) 0, we have:

f(0) -c

But we defined c f(0), which gives us:

c -c.

Therefore, 2c 0 implies c 0.

Thus, we conclude:

f(0) 0.

Step 4: The Continuity Argument

Now we can explore the implications further. The function f must map real numbers to real numbers continuously. If we assume f is continuous and f(f(x)) -x, this would imply that f is an odd function because:

f(-x) -f(x)

Thus:

f(-x) -f(x)

Step 5: Contradiction with the Intermediate Value Theorem

Given that f is continuous and odd, if we evaluate f at any point x_0 where f(x_0) is positive, then f(f(x_0)) -x_0 should yield a negative value.

By the Intermediate Value Theorem, since f is continuous, it must take every value between f(x_0) and f(-x_0). This is a contradiction because f(f(x_0)) should not jump from positive to negative without passing through zero.

Conclusion

Therefore, we arrive at a contradiction indicating that no such continuous function f can exist such that f(f(x)) -x for all x.