C^*-smooth

Let X be a continuum. Define $C^*: C(X) \to C(C(X))$ by C^*(A) = C(A). It is known that for any continuum X the function C^* is upper semi-continuous, [64, Theorem 15.2, p. 514], and it is continuous on a dense $G_\delta$ subset of C(X), [64, Corollary 15.3, p. 515]. A continuum X is said to be C^*-smooth at $A \in C(X)$ provided that the function C^* is continuous at A. A continuum X is said to be C^*-smooth provided that the function C^* is continuous on C(X), i.e., at each $A \in C(X)$ (see [64, Definition 5.15, p. 517]).