1. Why These Rules Matter
The algebra of continuous functions tells us how continuity behaves when functions are combined. If we already know some functions are continuous, these rules help us quickly conclude that new functions made from them are also continuous.
It saves time because we don’t have to check limits repeatedly.
2. Sum and Difference Rule
If two functions are continuous at a point, then their sum and difference are also continuous at that point.
2.1. Statement
\text{If } f \text{ and } g \text{ are continuous at } a, \text{ then }
(f + g)(a) = f(a) + g(a)
(f - g)(a) = f(a) - g(a)
3. Constant Multiple Rule
Multiplying a continuous function by a constant keeps it continuous.
3.1. Statement
\text{If } f \text{ is continuous at } a, \text{ then } c f(x) \text{ is also continuous at } a.
4. Product Rule
The product of two continuous functions is continuous at that point.
4.1. Statement
(fg)(a) = f(a) g(a)
5. Quotient Rule
A quotient of two continuous functions is continuous unless the denominator becomes zero.
5.1. Statement
\frac{f}{g} \text{ is continuous at } a \text{ if } g(a) \ne 0.
6. Composition Rule
The composition of continuous functions is also continuous. This is one of the most powerful rules.
6.1. Statement
\text{If } f \text{ is continuous at } b \text{ and } g \text{ is continuous at } a \text{ where } g(a) = b, \text{ then } (f \circ g)(x) \text{ is continuous at } a.
7. Examples Illustrating the Rules
These quick examples show how the rules are applied.
7.1. Example 1 — Sum Rule
f(x) = x^2, g(x) = 3x. Both are continuous everywhere, so:
f(x) + g(x) = x^2 + 3x
is continuous everywhere.
7.2. Example 2 — Product Rule
f(x) = x, g(x) = e^x. Each is continuous, so:
(fg)(x) = x e^x
is continuous everywhere.
7.3. Example 3 — Quotient Rule
f(x) = x^2, g(x) = x - 1. Then
\frac{x^2}{x - 1}
is continuous except at x = 1, where the denominator is 0.
7.4. Example 4 — Composition Rule
f(x) = x^3, g(x) = \sqrt{x}. Both continuous in their domains.
Then:
(f \circ g)(x) = (\sqrt{x})^3 = x^{3/2}
is continuous for x ≥ 0.
8. Why These Rules Are Helpful
These algebraic rules let us quickly build new continuous functions from known ones. This forms the basis for most functions used in calculus, making these rules essential for working smoothly with continuity.