Try graphing it out on your graphing calculator or a web-based grapher. You will see that it is continuous everywhere. However, at x=0 it comes to a sharp point.
"Differentiable"
When you think of differentiable, think... can I find the slope everywhere? Recall that when a function comes to a sharp point, you cannot find the slope there. This is because if you approach it from the left, you will get one slope, and if you approach it from the right, you will get the opposite (usually) or at least a very different slope.
Putting "differentiate the square root of the absolute value of X" into Wolfram Alpha, you will get this result:
This result is valid everywhere except at 0... which is excluded from the domain because the denominator will be 0, which is not allowed.
Wolfram Alpha will graph this for you also. You can see the graph has an asymptote at 0 and goes to positive infinity as you approach zero from the positive side, and negative infinity as you approach zero from the negative side. This is what we expected given the sharp point in the graph of the initial function.
Cheers!
--Louise
