Determine the value of x for which the function f(x) ={¨€(5x,if x<0;1,if x<0; -5x,if x>0)©Èis discontinuous.
To determine the value of x for which the function f(x) is discontinuous, we need to find the points at which the function exhibits a jump, a hole, or an infinite limit.
First, let's examine the given function: f(x) = { € (5x, if x < 0; 1, if x < 0; -5x, if x > 0).
The function is split into three pieces based on the value of x: for x < 0, for x = 0, and for x > 0.
Now, let's analyze each piece separately:
1. For x < 0: The function f(x) = 5x. It represents a line with a slope of 5. This part of the function is continuous since it is defined for all x < 0.
2. For x = 0: The function f(x) = 1. This means that the function takes the value 1 when x equals 0. This is a single point and does not contribute to any discontinuity.
3. For x > 0: The function f(x) = -5x. Similar to the first piece, this is a line with a slope of -5. It is also continuous since it is defined for all x > 0.
Therefore, the function f(x) is continuous for all values of x, except for x = 0. At x = 0, the function exhibits a jump from 5x to 1. Therefore, the value of x for which the function is discontinuous is x = 0.