#2 and #3

2. Let S be a continuous real or complex function defined on a topological space X, and assume that f is not identically zero, i.e., that the set Y {x : f(x) + 0} is non-empty. Prove in detail that the function 1/8 defined by (1/8)(x) = 1/[(x) is continuous at each point of the subspace Y. 3. Let X be a topological space and A a subalgebra of 6 (X, R) or 6 (X,C). Show that its closure A is also a subalgebra. If A is a subalgebra of 6 (X,C) which contains the conjugate of each of its functions, show that A also contains the conjugate of each of its functions. 2. Let S be a continuous real or complex function defined on a topological space X, and assume that f is not identically zero, i.e., that the set Y {x : f(x) + 0} is non-empty. Prove in detail that the function 1/8 defined by (1/8)(x) = 1/[(x) is continuous at each point of the subspace Y. 3. Let X be a topological space and A a subalgebra of 6 (X, R) or 6 (X,C). Show that its closure A is also a subalgebra. If A is a subalgebra of 6 (X,C) which contains the conjugate of each of its functions, show that A also contains the conjugate of each of its functions