Algebra of Continuous Functions

Just as the concept of limits follows specific algebraic rules, the continuity of functions does as well. Because continuity at a point is entirely determined by the limit of the function at that point, the algebra of continuous functions mirrors the algebra of limits.

These rules allow us to easily determine the continuity of complex functions by breaking them down into simpler, known continuous parts.

If f and g are two real functions that are continuous at a real number x = c, then the following algebraic combinations are also continuous at x = c:

• Addition: f + g

• Subtraction: f - g

• Multiplication: \[f \cdot g\]

• Division: \[\frac{f}{g}\] (provided that \[g(c) \neq 0\])

• Reciprocal of a Continuous Function: If g is continuous and g(x) ≠ 0, then \[ \frac{1}{g(x)} \]

• Rational Functions:  \[ f(x) = \frac{p(x)}{q(x)} \]
  where \[p(x)\] and \[q(x)\] are polynomial functions and \[ q(x) \neq 0. \]is continuous wherever it is defined.

• Trigonometric Functions: \[\sin x\] and \[\cos x\] are continuous for all real \[x\], while \[\tan x = \dfrac{\sin x}{\cos x}\] is continuous wherever \[\cos x \neq 0\], i.e. \[x \neq \dfrac{(2n+1)\pi}{2}\], \[n \in \mathbb{Z}\].

If g is continuous at c, and f is continuous at g(c), then their composite function \[(f \circ g)\], defined as \[(f \circ g)(x) = f(g(x))\], is also continuous at c.

Show that the function defined by f(x) = sin (x²) is a continuous function.

Solution: 

Let \[ g(x) = \sin x \]

and \[ h(x) = x^2. \]

Since both \[g\] and \[h\] are continuous functions,

\[ f(x) = (g \circ h)(x) = g(h(x)) = \sin(x^2) \]

is continuous for all real \[x\].

Show that the function f defined by f(x) = |1 – x + |x||, where x is any real number, is a continuous function.

Solution:

Let \[ g(x) = 1 - x + |x| \] and \[ h(x) = |x|. \]

Since polynomial functions and \[|x|\] are continuous, 

\[ g(x) \] is continuous.

Also \[h\] is continuous.

Therefore, \[ f(x) = h(g(x)) \] is continuous.

•  If \[f\] and \[g\] are continuous at \[c\], then \[f+g\], \[f-g\], and \[fg\] are continuous at \[c\].
• \[\dfrac{f}{g}\] is continuous provided 
  \[ g(c) \neq 0. \]
• If \[f\] is continuous, then \[\lambda f\] is continuous.
• If \[g\] is continuous and non-zero, then \[ \frac{1}{g} \] is continuous.
• Rational functions are continuous on their domains.
• sin x and cos x are continuous for all real \[x\].
• \[\tan x\] is continuous wherever \[\cos x \neq 0\].
•  If \[g\] is continuous at \[c\] and \[f\] is continuous at \[g(c)\], then \[ f \circ g \] is continuous at \[c\].