If a function f is continuous on the interval from negative infinity to positive infinity, we can make several key observations about its graph:
- The graph has no breaks or holes: Since f is continuous, its graph does not have any jumps, asymptotes, or holes. You can draw the graph of f without lifting your pencil.
- The function can be evaluated at every point on the real line: For every real number x, the function f(x) exists. This means we can find an output for every input.
- The graph may extend indefinitely in both directions: Continuity does not inherently restrict how far the graph can go. It can approach infinity in both the positive and negative directions.
- It may have local extrema: A continuous function can have peaks and valleys (local maxima and minima), as long as the function is defined on the entire interval.
These properties highlight the essential nature of continuity in the graph of a function over an infinite interval.