If f(x) is discontinuous, determine the reason.f(x) = {x^2 -4, x<1 { x+4, x>1 f(x) is continuous for all real numbers The limit as x approaches 1 does not exist f(1) does not equal the limit as x approaches 1 f(1) is not defined
To the left of 1 the function is a quadratic polynomial, to the right, it is a linear polynomial. Polynomial functions are always continuous, so the only candidate point for discontinuity is x=1.
The left limit is calculated with 1 substituted in [tex]x^2+4[/tex], which gives 5.
The right limit, is computed using the rule for the right part of 1, that is x+4.
Thus, the right limit is 1+4=5.
So, both left and right limits are equal. Now if f(1) is 5, then the function is continuous at 1.
But the function is not defined for x=1, that is x=1 is not in the domain of the function. Thus, we have a "whole" (a discontinuity) in the graph of the function.