Q1(k). Well-behaved functions: two tests

Answer in brief

A well-behaved function is single-valued, finite, continuous (with a continuous first derivative) and square-integrable. (i) is well-behaved; (ii) is not: it becomes infinite at x = 4.

Key points

• Well-behaved: single-valued, finite, continuous, continuous dϕ/dx, and ∫|ϕ|² dx finite (normalisable), because ϕ*ϕ dx is a probability.
• (i) e⁻ᵅˣ: one value at each x; lies between 1 and e⁻ᵅ; continuous, with dϕ/dx = −αe⁻ᵅˣ continuous.
• ∫e⁻²ᵅˣ dx is finite over 0 to α and even 0 to ∞ (1/2α) ⇒ (i) well-behaved.
• (ii) 1/(4 − x): x = 4 lies inside 1 < x < 10; ϕ → +∞ from the left, −∞ from the right ⇒ (ii) not well-behaved.