A function f is continuous at the point if the following conditions are satisfied.
1) is defined.
2) exists.
3) .

Determine the values of x, if any, at which the function is discontinuous. At each point of discontinuity, state the condition(s) for continuity that are violated.

A) ; conditions 1, 2
B) ; conditions 2, 3
C) ; conditions 1, 2
D) ; condition 2
E) Continuous everywhere.

Question

A function f is continuous at the point  if the following conditions are satisfied. ​
1)  is defined.
2)  exists.
3)  .
​
Determine the values of x, if any, at which the function is discontinuous. At each point of discontinuity, state the condition(s) for continuity that are violated.
​  ​  ​
A) ; conditions 1, 2 
B) ; conditions 2, 3 
C) ; conditions 1, 2 
D) ; condition 2 
E) Continuous everywhere.

Quizplus · Accepted Answer

For the function to be discontinuous, condition 3 must not be satisfied. Extra attention should be paid to points where the function is undefined (condition 1) or where the limit does not exist (condition 2).
At $x=4$, the function is undefined (condition 1) since the denominator of the fraction becomes zero.
At $x=6$, the limit does not exist (condition 2) since the left and right limits are not equal.
At $x=8$, the function is continuous since all three conditions are met.
At $x=9$, the limit does not exist (condition 2) since the left and right limits are not equal.
Therefore, the only point(s) of discontinuity are $x=4$ and $x=9$ where condition 3 is violated. However, only choice B correctly identifies both points and the conditions that are not satisfied.

A Function F Is Continuous at the Point If