## 2011-2012

Problem 2: Continuity

Let *f*(*x*) be the function defined below:

Determine whether *f*(*x*) is continuous at *x* = 0 and explain your answer.

Note: You may use a graphing calculator to examine the graph of *f*(*x*).

### Solution:

If *f*(*x*) if continuous at *x* = 0, its left- and right-hand limits exist at *x* = 0, and they are both equal to *f*(0). Consider the graph of the function below.

This sine curve is a "damped" function; it is already zoomed in quite far, but feel free to zoom in to your heart's content. The function will wriggle its way to a height of 0 as you approach the *y*-axis from the right and from the left. Therefore, the general limit exists, and it is equal to 0.

According to the piecewise-defined function, *f*(0) = 0. (It's a good thing, too, because substituting 0 into *x*2sin(1/*x*) would have been a deal-breaker. You're not allowed to have a 0 in a denominator.)

Because the limit of *f*(*x*) exists as *x* approaches 0 and it equals *f*(0), you conclude that *f*(*x*) is continuous at 0.