Rule 1: g continuous, f continuous => f(g(x)) continuous The composition of continuous functions is continuous.

Rule 2: g discontinuous, f continuous => f(g(x)) may or may not be continuous Example: g(x)=[x] (discontinuous), f(x)=$x^2$ (continuous). f(g(x))=[x]^2 is discontinuous at integers.

Rule 3: [g(x)] discontinuities [g(x)] is discontinuous where g(x) crosses an integer value. Find all x where g(x) = n (integer) and check if g actually crosses through n.

Rule 4: Composition with |f| |g(x)| is always continuous if g is continuous. Differentiability fails at zeros of g with nonzero derivative.

JEE Favorites:

• [sin x]: discontinuous where sin x = 0, +/-1
• [$x^2$]: discontinuous where $x^2$ is a positive integer
• sin([x]): discontinuous at all integers (since [x] jumps)
• [2sin x]: discontinuous where 2sin x crosses integers