Define inferior semicontinuity function and show that f is continuous at a point if, and only if, it is upper and lower semicontinuous at that poin...

Define inferior semicontinuity function and show that f is continuous at a point if, and only if, it is upper and lower semicontinuous at that point.
a) Define inferior semicontinuity function and show that f is continuous at a point if, and only if, it is upper and lower semicontinuous at that point.
b) Prove that a subset A ⊂ R is open if, and only if, its characteristic function ξA: R → R (defined by ξA(x) = 1 if x ∈ A and ξA(x) = 0 if x ∉ A) is inferior semicontinuous.
c) State and prove a similar result for closed sets.
d) Show, more generally, that for every subset X ⊂ R, its characteristic function ξX: R → R is discontinuous precisely at the boundary points of X. Given fr X, show that ξX is upper semicontinuous at a if X and lower semicontinuous if a / X. Conclude that the function f: R → R, defined by f(x) = 1 for x ∈ Q and f(x) = 0 for x irrational, is upper semicontinuous at rational numbers and lower semicontinuous at irrational numbers.
e) Let f: R → R be defined by f(x) = sin(1/x) if x ≠ 0 and f(0) = c. Show that f is upper semicontinuous at the point 0 if, and only if, c ≥ 1. (And lower semicontinuous if, and only if, c ≤ 1.) Taking 1 < c < 1, show that f is not lower or upper semicontinuous at the point 0.
f) The functions f and g: R → R, where f(0) = g(0) = 0 and, for x ≠ 0, f(x) = x sin(1/x), g(x) = 1/|x|, are inferior semicontinuous, but their product f · g is not a semicontinuous function at the point 0.
g) For f: X → R to be upper semicontinuous at the point a ∈ X ∩ X′ is necessary and sufficient that limx→a sup f(x) ≤ f(a). The analogous result holds for lower semicontinuity.
h) The sum of two upper semicontinuous functions at a point still has the same property. Use item (e) with c = 1 and c = 1 to give an example of two semicontinuous functions (one upper and one lower) whose sum is not semicontinuous. Show that if f is upper semicontinuous and g is lower semicontinuous, then f + g is not necessarily semicontinuous.