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.