Properties of Continuous Functions

Continuous functions exhibit smooth, uninterrupted behavior, and combining them through standard operations retains this continuity. If f and g are continuous at a point a, then the functions f+g, f-g, cf (where c is a constant), fg, and fg (provided g(a)a) are also continuous at a. This allows the construction of complex functions from simpler continuous parts without losing smoothness.

Polynomials, which are expressions formed by sums of powers of x with constant coefficients, are continuous for all real numbers. That is, they are continuous across the entire set of real numbers from negative infinity to positive infinity. Rational functions, which are ratios of two polynomials, are continuous at all points where the denominator is nonzero.

Standard functions such as sin x, cos x, ex, ln x , and inverse trigonometric functions are continuous throughout their defined domains. For instance, ln x is continuous on the interval (0, ∞), while sin x and cos x are continuous for all real values of x.

Moreover, if g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is also continuous at a, preserving continuity through nested operations.