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.