Can two different functions have an infinite number of solutions?

[Tim_B]

Let f(x) and g(x) be non-piecewise defined functions that are defined for all real numbers. Furthermore, let f(x) and g(x) be continuous and differentiable at all points.
Are there two functions f(x) and g(x) such that f(x)=g(x) for all points over some interval (a,b], and f(x)≠g(x) for all points over some interval (b,c)? Assume a≠b and b≠c.
Basically, what I'm asking is this: can two different functions equal one another for all points over some interval? If I'm not making myself clear, see the attached picture:
https://www.physicsforums.com/attachment.php?attachmentid=59652&stc=1&d=1371520767
Thanks for your help. PS: This is my first post.

 

[Office_Shredder]

Non-piecewise defined isn't a very clearly defined term. For example is |x| allowed? What about f(x) = x if x > 0, and -x if x<0?

 

[Simon Bridge]

Welcome to PF;

Basically, what I'm asking is this: can two different functions equal one another for all points over some interval?

... yes they can. Unless you tighten your definition to the point where they can't.

 

[Tim_B]

Non-piecewise defined isn't a very clearly defined term. For example is |x| allowed? What about f(x) = x if x > 0, and -x if x<0?

I realize that, but |x| isn't differentiable over its entire domain anyway.

 

[Tim_B]

Welcome to PF;
... yes they can. Unless you tighten your definition to the point where they can't.

Could you give an example or two?

 

[Office_Shredder]

I realize that, but |x| isn't differentiable over its entire domain anyway.

Yeah but |x³| is.

Whether it's differentiable or not is totally irrelevant anyway. Your question as posed doesn't make sense - saying "not piecewise defined" is not a well-defined statement.

 

[HallsofIvy]

Once you have said "differentiable for all x" you don't need "not piecewise defined".

No, the fact that two functions are equal at every point on an interval does not mean they are equal for other points.

 

[Tim_B]

Yeah but |x³| is.

Whether it's differentiable or not is totally irrelevant anyway. Your question as posed doesn't make sense - saying "not piecewise defined" is not a well-defined statement.

Good point. I should redefine my terms.