# Homotopy and the continuity of the solution

Consider two homotopic functions f(x) and g(x) and H(x,t) being the path function where t belongs to the interval [0,1]. Suppose that the root of H(x,t) with respect to x is x(t). Is function x(t) continous in [0,1]? Gracias!

P.S. The reason I interest in this question is that when operating comparative static analysis, for instance, F(X,t) = 0 where F and X may be vectors of same dimensions and t is exogenous parameter, conducting analysis in the neighborhood of t = 0 will considerably simplify the process. However, applying the result where t = 0 to the condition where t nearly equals zero requires the solution of X(t) is continous at the point t = 0.

@stepan-a Do you have an idea?

Hi, Obviously the answer is problem specific and depends on the properties of function H(x,t). You will find detailed explanations about the existence and properties of the solution path in the textbook by Zangwill and Garcia (1981, Pathways to solutions, fixed points and equilibria, Prentice-Hall Series in Computational Mathematics).

Best,
Stéphane.

Forgot to mention that the book is available here:

Best,
Stéphane.

Thank you very much! It’s very of you and your answer really helps me a lot!