Interiors of continuous images of self-similar sets with overlaps

Let K be the attractor of the following iterated function system {S-1 (x) = lambda(x), S-2(x) = lambda(x) + c - lambda, S-3(x) = lambda(x) + 1 - lambda}, where S-1 (I) boolean AND S-2(I) not equal empty set (S-1(I) boolean OR S-2(I)) boolean AND S-3(I) = empty set, and I = [0,1] is the convex hull of K. Let d(1) = 1-c-lambda/lambda < 1/1-c-lambda = d(2). Suppose that f is a continuous function defined on an open set U subset of R-2. Denote the image f(U) (K, K) = {f (x,y) : (x,y) is an element of (K x K) boolean AND U}. If partial derivative(x)f, partial derivative(y)f are continuous on U, and there is a point (x(0), y(0)) is an element of (K x K) boolean AND U such that vertical bar partial derivative(y)f vertical bar((x0,y0))/partial derivative(x)f vertical bar ((x0,y0))vertical bar is an element of (d(1,)d(2)) or vertical bar partial derivative(x)f vertical bar((x0,y0))/partial derivative(y)f vertical bar((x0,y0))vertical bar is an element of (d(1,)d(2)), then f(U) (K, K) contains an interval. As a result, we let c = lambda = 1/3, and if f(x,y) = x(alpha)y(beta)), x (alpha beta not equal 0), x(alpha)+/- y(alpha) (alpha not equal 0) sin(x) cos(y), or x sin(xy), then f(U)(C, C) contains an interval, where C is the middle-third Cantor set.