1. Why Standard Functions Are Usually Continuous
Most commonly used functions show smooth behaviour within their domains. This makes them automatically continuous unless something in their definition creates a break, hole, or division by zero.
Knowing which functions are always continuous helps avoid checking continuity from scratch each time.
2. Constant and Identity Functions
Both constant and identity functions behave smoothly everywhere in their domains.
2.1. Constant Function
f(x) = c is continuous for all x.
2.2. Identity Function
f(x) = x is continuous for all x.
3. Polynomial Functions
Every polynomial is continuous everywhere. Their graphs have no breaks and are smooth throughout.
3.1. Example
f(x) = 3x^2 - 5x + 7
This function is continuous for all real values of x.
4. Rational Functions
Rational functions are continuous everywhere except where the denominator becomes zero.
4.1. Example
f(x) = \frac{x^2 - 1}{x - 1}
Continuous for all x except x = 1.
5. Root Functions
Functions involving square roots, cube roots, and other fractional powers are continuous on their valid domain.
5.1. Square Root Example
f(x) = \sqrt{x}
Continuous for x ≥ 0.
5.2. Cube Root Example
f(x) = \sqrt[3]{x}
Continuous for all x.
6. Exponential and Logarithmic Functions
These functions are continuous wherever they are defined.
6.1. Exponential Function
f(x) = e^x
Continuous for all real x.
6.2. Logarithmic Function
f(x) = \ln x
Continuous for x > 0.
7. Trigonometric Functions
Basic trigonometric functions are continuous on their respective domains.
7.1. Examples
- sin x — continuous for all real x
- cos x — continuous for all real x
- tan x — continuous except at odd multiples of \( \frac{\pi}{2} \)
8. Inverse Trigonometric Functions
Inverse trigonometric functions are continuous on their principal domains.
8.1. Examples
- arcsin x — continuous for x ∈ [-1, 1]
- arccos x — continuous for x ∈ [-1, 1]
- arctan x — continuous for all real x
9. Piecewise Functions
Piecewise functions are continuous on each piece of their domain. To check continuity at boundary points, the left-hand and right-hand limits must match the value of the function at that boundary.
9.1. Example
f(x) = \begin{cases} x + 1, & x < 2 \\ 3x, & x \ge 2 \end{cases}
Check continuity at x = 2 by comparing both one-sided limits.
10. Summary
Most standard functions are continuous in their domains. Breaks appear only when something in the definition fails — such as a denominator becoming zero, a negative number under an even root, or a piecewise function with unmatched values.