To show that , you need to show that the difference can be made arbitrarily small, provided that is small enough. To get something "arbitrarily small", you usually try to make it smaller than some given . But we already have an here, so we had better choose another letter, say . Let's try to make
where is given.
The first thing to notice is that (because when you integrate the constant over a ball, you get that constant times the volume of the ball. Therefore
Next, the absolute value of an integral is always less than the integral of the absolute value (of the function in the integral). Therefore
Now, notice that if we could arrange that for all , then that last integral would be less than , as required. But that condition, for all , follows from the given fact that is continuous at (can you see why?).