Continuous Functions in Calculus: Definition, Properties, and Examples

Explore the concept of continuous functions in calculus. This guide provides a formal definition of continuity, explains the conditions for continuity at a point and over an interval, and illustrates continuity with various examples of common functions.

Continuous Functions in Discrete Mathematics

What is a Continuous Function?

A continuous function is a function whose graph is unbroken—you can draw the entire graph without lifting your pen. More formally, a function is continuous at a point if its value at that point equals the limit of the function as x approaches that point from both the left and the right. Intuitively, this means that there are no jumps or breaks in the graph at that point. This is a fundamental concept in calculus and analysis, and understanding continuity is crucial for many mathematical and computational tasks.

Definition of Continuity at a Point

A function f(x) is continuous at a point x = a if all three of these conditions are met:

  1. f(a) exists (the function is defined at x = a).
  2. limx→a f(x) exists (the limit of the function as x approaches a exists).
  3. limx→a f(x) = f(a) (the limit equals the function's value at that point).

Continuity Over an Interval

A function is continuous over an interval if it's continuous at every point within that interval. The graph of such a function is a smooth, unbroken curve throughout the interval.

Examples of Continuous Functions

Many common functions are continuous over their domains (the set of all valid input values):

  • Polynomial functions (e.g., f(x) = x² + 2x + 1)
  • Trigonometric functions (sin(x), cos(x), tan(x))
  • Exponential functions (ex)

Properties of Continuous Functions

If f(x) and g(x) are continuous at x = a:

  • f(x) + g(x) is continuous at x = a.
  • f(x) - g(x) is continuous at x = a.
  • f(x) * g(x) is continuous at x = a.
  • f(x) / g(x) is continuous at x = a, provided g(a) ≠ 0.
  • If f(x) is continuous at g(a), then f(g(x)) is continuous at x = a (composition of functions).

Important Theorems on Continuous Functions

  • All polynomial functions are continuous everywhere.
  • sin(x), cos(x), ex, and arctan(x) are continuous for all real numbers.
  • If f(x) and g(x) are continuous on an interval [a, b], then f(x) + g(x), f(x) - g(x), and f(x) * g(x) are continuous on [a, b].
  • Rational functions (ratios of polynomials) are continuous except where the denominator is zero.

Discontinuous Functions

A function that is not continuous is called discontinuous. Types of discontinuities include:

  • Jump Discontinuity: The left-hand and right-hand limits at a point are different.
  • Removable Discontinuity: The limit exists, but it doesn't equal the function's value at that point.
  • Infinite Discontinuity: The function approaches positive or negative infinity at a point.

Examples: Continuity and Discontinuity

Example 1: Continuity at a Point

Is f(x) = 5x - 8 continuous at x = 5?

Solution: limx→5 f(x) = 17; f(5) = 17. Therefore, it's continuous at x = 5.

Example 2: Discontinuity at a Point

Is the following function continuous at x = 5?

f(x) = x - 7  for x < 5
f(x) = 6      for x ≥ 5

Solution: limx→5⁻ f(x) = -2; limx→5⁺ f(x) = 6. The limits are different; there's a jump discontinuity.

Example 3: Finding Conditions for Continuity

Find the relationship between a and b for the function f(x) to be continuous at x = 4:

f(x) = ax - 7  for x ≤ 4
f(x) = bx + 6  for x > 4

Solution: limx→4⁻ f(x) = limx→4⁺ f(x) = f(4) implies 4a - 7 = 4b + 6, so 4a - 4b = 13.

Conclusion

Understanding continuity is fundamental in calculus and analysis. Knowing how to identify continuous functions and their properties is crucial for solving various mathematical problems.