Determine the values of x, if any, at which each function is discontinuous. At each number where f is discontinuous, state the condition(s) for continuity that are violated.
f(x) = x+5 if x<0
= 2 if x=0
= -x^2+5 if x>0
To determine the values of x at which the function f(x) is discontinuous, we need to look for any points in the domain where there are abrupt changes or breaks in the function.
In this case, we need to consider the three cases defined by the function:
1. For x < 0: f(x) = x + 5
2. For x = 0: f(x) = 2
3. For x > 0: f(x) = -x^2 + 5
Let's analyze each case to find any points of discontinuity:
1. For x < 0: The function f(x) = x + 5 is a linear equation, which is continuous for all real values of x. There are no discontinuities in this case.
2. For x = 0: The function f(x) = 2 is a constant function with a single value at x = 0. As the function is defined only at this specific point, it is also continuous. No discontinuities exist at x = 0.
3. For x > 0: The function f(x) = -x^2 + 5 is a quadratic equation, which is also continuous for all real values of x. There are no abrupt changes or breaks in the function in this case.
In conclusion, the function f(x) = x + 5, f(x) = 2, and f(x) = -x^2 + 5 are all continuous for all values of x, and there are no points of discontinuity.
Therefore, there are no values of x at which the function f(x) is discontinuous, and no conditions for continuity are violated.