Labware - MA35 Multivariable Calculus - Single Variable Calculus

According to the epsilon-delta definition, a function of one variable f(x) is said to be continuous at x₀ if for any ε > 0 there exists a &delta such that | f(x) - f(x₀) | < ε whenever | x - x₀ | < &delta.

Demos

Continuity

The demo provides an illustration of how the epsilon-delta definition of continuity works. The function f(x) and its domain are specified in the control panel. The graph of f(x) appears in the window labeled "Graph: f(x)".

Along the x-axis domain are two red hotspots: ones lets you choose the point x₀ at which you want to test for continuity; the second lets you choose the size of a delta neighborhood of x₀. This neighborhood of x₀ and its image on the graph are both shown in magenta.

To test for continuity, start by choosing an epsilon in the control panel. This determines the location of the two blue horizontal bars which lie a distance epsilon above and below the point (x₀,f(x₀)).
The challenge is to find a small enough delta neighborhood of x₀ in the domain such that the image of that neighborhood lies in between the two blue bars. The function f(x) is called continuous at x₀ if it is always possible to meet this challenge for any value of epsilon.

Exercises

1. Consider the function f(x) = (x - 1)² for x ≠ 1 and f(1) = 2. Why is this function not continuous at x = 1? What if f(1) = k for some other constant?

2. Consider the function f(x) = tan(x - 1) for x ≠ 1 and f(1) = 2. Why is this function not continuous at x = 1? What if f(1) = k for some other constant?

3. Consider the function f(x) = sin(1/(x-1)) for x ≠ 1 and f(1) = 2. Why is this function not continuous at x = 1? What if f(1) = k for some other constant?