Page 15 - Calc for the Clueless

Let's graph this function.

At x = 1, the limit from the left of f(x) = 1 is 1. The limit from the right of f(x) = x is 1. So

exists and equals 1. Part 1 of the definition is satisfied. But f(1) = 6. The function is not continuous at 1. (See
the jump to y = 6 at x = 1.)

At x = 3, the limits from the left and the right at 3 equal 3. In addition, f(3) = 3. The function is continuous at x
= 3. (Notice, no break at x = 3.)

At x = 6, the limit as x goes to 6 from the left is 0. The limit as x goes to 6 from the right is 4. Since the two
are different, the limit does not exist. The function is not continuous at 6 (see the jump). We do not have to
test the second part of the definition since part 1 fails.