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

The function is 

.

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 , 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.

The reason is now clear:

 f(1) is not defined