We have said that the function y = f(x) is continuous at point a, if f is defined at this point (f(a) exists) then the limit of this function exists when x tends to a, and this limit is the same as f(a). The following definitions are equivalent to this one and are more frequently used:

• f is continuous at a if
• f is continuous at a if

• f is continuous at a if f(a) ==

10.- Look carefully at the function. Bring P closer to A by dragging it with the mouse or by choosing values for the parameter (a+h) which are close to (a). You can see that h tends to 0 to both the left and right of a. What are the values of the increment of the function f(a+h) - f(a)? Explain your answer.

We saw earlier that a function, y = f(x), is continuous at point a of the domain if. Using the definition of the limit of a function at a set point we can say that:

"For any neighbourhood where f(a) is the centre and ß the radius, there is a neighbourhood with centre a and radius ð so that all its x points have their image, f(x), inside the neighbourhood with centre f(a) and radius ß".

In other words: "If x is less than ð away from a, its image f(x) is less than ß away from f(a)".

f(6) = 3.6

If ß = 1, the P points whose x-coordinate is located within the neighbourhood (6-ð , 6+ð ) = (5,22 , 6,78) has its image f(x) inside the neighbourhood with centre f(6) = 3.6 and radius 1. Check that this is true by dragging P over the curve with the mouse.

You can see that as x approaches 6 its image f(x) approaches f(6).

Change the value of the parameter ß in the window. You will see that for each value of ß, ð has a different value. Drag P with the mouse and look carefully at the values of x when its image is located in the interval (f(6)-ß, f(6)+ß) .

For any value we choose for parameter ß, we always get a value for radius ð, so that if x is less than ð away from 6, then f(x) is less than ß away from f(6) = 3.6 .

This means that

Let's continue working with the point a = 6.

12.- What is the value of the radius ð if ß = 0.8?

13.- For which values for x is the image f(x) less than 0.5 away from 3.6? You can find the answer by dragging P with the mouse over the curve. Write down your answer.

14.- In your exercise book work out the value of ð depending on ß when a = 6. Is the function continuous at the point x = 6?

Change parameter a to see the continuity of the function at other points on the graph in the window.

Select a = 5 and answer the following questions:

15.- For which values of x is its image is less than 2 away from f(5)?

16.- When a = 5, does a ð exist for each value of ß ? Work it out and write the answer in your exercise book. Is the function continuous at the point x = 5?