question

f(x) is differentiable everywhere

there are no cusps or corners -- no sharp turns/curves anywhere on the graph

(cusp on the left, corner on the right)

no vertical tangents anywhere, there's a horizontal tangent though at x=0

there are also no discontinutities since the function is continuous everywhere

Guest is probably right but I do not think you can really tell just by looking at this graph.

I am not familiar with the terms cusps and corners.

It looks like a piecemeal graph.  So its ability to be differentiable at the joining points would depend on exactly what the pieces are.

When is a graph differentiable, the definition from Google:

"A function is differentiable at a point when there's a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right."