Answer to Question #183584 in Economics for Ann

Question

Using relevant illustrations discuss the limit and continuity of functions

Answer 1

The function y = f (x) is called continuous at the point x = a, if the limit of the function y = f (x) as x tends to a, is equal to the value of the function at the point x = a.

The function y = f (x) is called continuous on the interval X if it is continuous at each point of the interval.

Let's use the constructed graphs of functions. In all three cases, the same curve is shown, however, these are three different functions.

Let's answer a few questions regarding these functions.

How do they differ from each other?

They differ from each other in their behavior at the point x = a.

How does the function behave at the point x = a on the first graph?

For the function y = f (x) with x = a, the value of the function does not exist, the function at the specified point is not defined.

How does the function behave at the point x = a on the second graph?

For the function y = f (x) with x = a, the value of the function exists, but it differs from the natural value of the function at the specified point.

How does the function behave at the point x = a on the third graph?

For the function y = f (x) with x = a, the value of the function exists, and it is equal to the natural value of the function at the specified point, that is, b.

If we exclude the point x = a from consideration, then all three functions will be identical.

In general, this entry looks like this:

$$lim_{x\rightarrow a} f(x)=b$$