HappyN said:
tiny-tim said:Find a function continuous at the origin only, and then repeat it.
HappyN said:
Landau said:
HappyN said:
HappyN said:
A function that is continuous on Z only means that it is only continuous at integer values of x, and is not continuous at any non-integer values. This means that the function has a "jump" or "break" in its graph at non-integer values.
To determine if a function is continuous on Z only, you can visually inspect its graph and look for any "jumps" or "breaks" at non-integer values. You can also use the epsilon-delta definition of continuity to prove that the function is not continuous at non-integer values.
Yes, a function can still be considered "good" or "useful" even if it is only continuous on Z. These types of functions are often used in discrete mathematics and computer science, and can still provide valuable information and insights.
One example of a function that is continuous on Z only is the floor function, denoted as f(x) = ⌊x⌋. This function returns the largest integer less than or equal to x, and is only continuous at integer values of x. At non-integer values, there is a "jump" or "break" in the graph.
A function that is continuous on Z only is different from a function that is continuous on all real numbers because it is not continuous at non-integer values. A function that is continuous on all real numbers has a smooth and unbroken graph, while a function that is continuous on Z only has "jumps" or "breaks" at non-integer values.