Continuity of Composite Functions: Explained with Examples

Functions are often built in layers: first compute an inner value, then feed it into an outer function. This layering is called composition, written (f ∘ g)(x) = f(g(x)). A natural question is whether chaining two continuous functions keeps the result continuous. The answer is yes, under one condition: the outer function must be continuous at the point where the inner function lands.

This matters because composition is everywhere in applied mathematics. A temperature reading passed through a calibration curve, a signal filtered twice, or the expression e^(sin x) are all compositions. The continuity theorem for compositions lets us evaluate their limits by simple substitution, working from the inside out.

Key Concepts

• Composition: (f ∘ g)(x) means apply g first, then apply f to the result.
• Continuity of compositions: if g is continuous at a and f is continuous at g(a), then f ∘ g is continuous at a.
• Inside-out evaluation: limits of compositions can be computed by evaluating the inner limit first, then applying the continuous outer function.
• Domain care: the inner function's outputs must land inside the outer function's domain.

Formulas

• (f ∘ g)(x) = f(g(x)).
• lim(x → a) f(g(x)) = f(lim(x → a) g(x)) = f(g(a)), when g is continuous at a and f is continuous at g(a).
• Example pattern: eˣ, sin x, and polynomials are continuous everywhere, so compositions built from them are continuous wherever the inner pieces are defined.

Worked Example

Show that h(x) = e^(x² − 3x) is continuous at x = 2, and find h(2).

Split h into an inner and outer layer. The inner function g(x) = x² − 3x is a polynomial, so it is continuous everywhere, including at x = 2. Its value there is g(2) = 4 − 6 = −2. The outer function f(u) = e^u is continuous at every real number, in particular at u = −2. Both conditions of the theorem hold, so h = f ∘ g is continuous at x = 2. By continuity, h(2) = e^(−2) ≈ 0.1353. A nearby check confirms smooth behavior: at x = 1.999, g(x) = 3.996001 − 5.997 = −2.000999, and e^(−2.000999) ≈ 0.1352, essentially the same as h(2). No jump, no hole.

Common Misconceptions

• Chaining continuous functions always works. The outer function must be continuous specifically at the inner function's output value, not just somewhere.
• You can always substitute into a composition. Substitution through the layers is valid only when the outer function is continuous at the inner limit.
• Composition is the same as multiplication. It is not. f(g(x)) nests the functions; f(x) · g(x) multiplies their outputs. The rules are completely different.

Real-World Applications

• Nested physical models: radioactive decay e^(−kt) composes the exponential with a linear time function.
• Signal chains: audio passing through an amplifier then a filter is a composition of two continuous transformations.
• Calibration: a raw sensor reading transformed by a calibration curve is a continuous composition.
• Finance: compound growth formulas compose exponentials with time-dependent rates.

In short, continuous layers stack safely: if each function is continuous where it needs to be, their composition is continuous too. Work from the inside out — check the inner function at the point, check the outer function at the inner value — and limits of compositions reduce to plain substitution.