Properties of Continuous Functions

A function is continuous over a domain if its graph forms a smooth, unbroken curve without gaps or sudden jumps. This means the function behaves predictably at every point, with no abrupt changes in value. Polynomials and standard functions like sin x and cos x are continuous across all real numbers, while rational functions remain continuous wherever their denominators are nonzero.

When continuous functions are combined through addition, subtraction, multiplication, or scaling by a constant, the resulting function remains continuous. These operations cannot create holes or jumps in the graph. For example, if f and g are continuous at point a, then f+g, f-g, and cf (where c is constant) are also continuous at a.

Division of continuous functions can introduce discontinuities if the denominator equals zero at any point. When one continuous function is divided by another, continuity is preserved only where the denominator remains nonzero. For instance, a ratio representing flow per unit area stays continuous because the denominator magnitude is always positive and never zero.

Rational functions, which are ratios of two polynomials, are continuous at all points where the denominator is nonzero. Since polynomials are continuous across all real numbers, rational functions inherit this property except at values that make the denominator zero. This makes them predictable and smooth everywhere they are defined.

If g is continuous at point a and f is continuous at g(a), then the composite function f(g(x)) is also continuous at a. This preserves continuity through nested operations, allowing you to build complex continuous functions from simpler continuous parts without losing smoothness or introducing discontinuities.

Standard functions such as sin x, cos x, e^x, ln x, and inverse trigonometric functions are continuous throughout their defined domains. For example, ln x is continuous on the interval (0, ∞), while sin x and cos x are continuous for all real values of x, making them reliable for direct substitution when evaluating limits.

Continuous functions satisfy the intermediate value theorem, which guarantees that if a function is continuous on a closed interval, it attains every value between its endpoints. This property is fundamental to understanding how continuous functions behave and ensures no values are skipped, reinforcing the concept of an unbroken graph.