This function is continuous at the origin; many books prove this as an application of the Squeeze Theorem. Because , it follows that . The Squeeze Theorem says
Hence , so is continuous at .However, is not differentiable at . We can verify this using the limit definition of the derivative:
which does not exist. You can see this in the applet on the left. Move your mouse over the picture to the left to start the animation; you may also have to click on the picture before it starts.