Can a Continuous Function Map One Value to Two Different Points?

[za10]

Homework Statement

Show that a continuous function such that for all c in the reals, the equation f(x) = c cannot have two solutions

Homework Equations

The Attempt at a Solution

I was thinking along the lines of a contradiction or somehow using intermediate value theorem but it seems like it is so easy it is hard.

 

[Office_Shredder]

What are you actually trying to prove here? Your sentence doesn't parse properly

 

[za10]

show that a continuous function cannot have two solutions for the equation f(x) = c for every c.

 

[Office_Shredder]

I have f(x) = c... am I solving for x given c? What you're trying to say is that f can't be two to one (i.e. for every point p in the image, there are two points in the preimage of p).

Contradiction is a good place to start. There have to be two points that f maps to zero, consider f on the interval between them