Essential Knowledge 1.2A1
Essential Knowledge 1.2A1 Students will know that a function f is continuous at x = c provided that f(c) exists, exists, and .
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Mathematicians use limits to define continuity like so:
A function f(x) is infinite at point c if
According to this definition, there are three ways a function can be discontinuous at a point. A function is continuous if
f(c) exists
exists
Let's look at examples for these three methods of failure
Example 1
Is continuous at x = 0?
Let's try the first rule
1. f(c) exists
Well,
, and since this is not a limit, this is undefined. Therefore, the function is not continuous at x = 0 based off of the first rule of continuity. (It would also fail the other two)
Example 2
For the function shown in the graph below, is the function continuous at x = 4?
They didn't give us a function for this graph, so we just have to use the picture. (We could figure out a function if we really wanted to but the picture will be more than enough.)
Let's try the first rule
1. f(c) exists
Well, it does exist. The shaded blue dot means that when x = 4, y = 3. The point at (4, 2) is hollow to show it is not part of the curve. f(4) = 3, so the first rule is not broken
Let's try the second rule
2. exists
and therefore the limit does not exist. Therefore the function is not continuous at x = 4 according to the second rule of continuity
Example 3
For the function shown in the graph below, is the function continuous at x = 4?
Let's try the first rule
1. f(c) exists
Well, it does exist. The shaded blue dot means that when x = 4, y = 1. The point at (4, 3) is hollow to show it is not part of the curve. f(4) = 1, so the first rule is not broken
Let's try the second rule
2. exists
and therefore the limit exists.
Let's try the third rule
3.
and f(4) = 1 therefore and the limit does not exist
Example 4
For the function shown in the graph below, is the function continuous at x = 4?
This one finally passes all continuity tests, so the function is continuous at x = 4