Continuity at a Point via Formulas Examples

Determine whether the function 

is continuous at x = 1.

To begin, we should make sure the function is defined at x = 1. It is, and f(1)=1. 

Now we need to check on to see if it exists. As x approaches 1 from the left, we find 

Similarly, as x approaches 1 from the right we find 

Since the one-sided limits agree, 

Since , f is continuous at x = 1.

Determine whether the function 

 is continuous at x = 2.

When we try to evaluate the function at x = 2, we find
 

which is undefined. Since f(2) doesn't exist, the function can't be continuous at x = 2.

Determine whether the function 

is continuous at x = 0.

When we evaluate the function at 0 we find 
 
which is fine. We will see if the limit of f exists as x approaches 0. As x approaches 0 from the left, x will be less than 0. We use the first part of the function definition and find

As x approaches 0 from the right, x will be greater than 0. We need to use the second line of the function definition. We find

Since the one-sided limits do not agree,  does not exist, and the function is not continuous at x = 0.

Determine whether the function 

is continuous at x = 0.

This function can't be continuous at x = 0, since the the function isn't even defined there. The function is defined for x < 0 and x > 0, but not for x = 0.

To add insult to injury, doesn't exist since 
    
.

At what values is f discontinuous? 

The function f will be discontinuous at any value of c where f(c) is undefined, where  does not exist, or where . Think of such values as problem spots. We need to look at the function's definition and find these "problem spots".

Where is f undefined?

It looks like x = 2 could be trouble, since then  

However, when x = 2 we're not using this part of the function's definition. We only use  when x < 0. Since 2 > 1, we have f(2) = 4.

The function f is undefined at . When  we're using the piece of the function definition that says

so

Therefore f is discontinuous at .

Where can  not exist?

This can happen in between the "pieces". That is, where the function definition changes,  or where individual pieces have undefined limits. The only place an individual piece of this function has an undefined limit is at , which we've already taken care of.

For this function, the function definition changes at 0 and 1. We need to investigate the limit of f as x approaches each of these values. First off, does  exist? For the left-sided limit, we find

For the right-sided limit, we find

Since the one-sided limits disagree,  does not exist, and f is discontinuous at x = 0.

Does  exist?

For the left-sided limit, we find

For the right-sided limit, we find

Since the one-sided limits agree, 

Finally, we need to see if f(c) and  disagree. Again, we only need to inspect those values of c where the function's definition changes over. Since  the function f is continuous at 1.

We have no more possible trouble spots, f is discontinuous only at  and x = 0.