or each of the following functions R → R, say if it is continuous in the cofinite topology.You do not need to justify your answer.(a) f (x) = x2 + 2x + 5 [2,U]It is continuous because |f −1(a)| ≤ 2 for all a ∈ R.(b) g(x) = sin(x + 2) [2,U]It is not continuous because |g−1(0)| = ∞.(c) h(x) = e3x [2,U]It is continuous because |h−1(a)| ≤ 1 for all a ∈ R

The cofinite topology on R is the topology where the open sets are the empty set and the sets whose complements in R are finite.

(a) f (x) = x^2 + 2x + 5: This function is continuous in the cofinite topology. In the cofinite topology, every function is continuous because the preimage of an open set (which is either the empty set or a cofinite set) is always open (either empty or cofinite).

(b) g(x) = sin(x + 2): This function is also continuous in the cofinite topology. As mentioned above, every function is continuous in the cofinite topology.

(c) h(x) = e^3x: This function is continuous in the cofinite topology. Again, every function is continuous in the cofinite topology.

The given justifications about the cardinality of the preimage of a point do not apply here, because they do not relate to the definition of continuity in the cofinite topology.