As near as I can tell the graphs seem to be mirrored around the origin - that is, they're graphs of abs(x), not x. Which naturally causes a discontinuity for any function whose slope isn't zero at the origin...
I think that you mean they are graphs of x ↦ f(abs(x)), not just f; and that this causes a point of non-differentiability, not necessarily a discontinuity. Of course, the even extension x ↦ f(abs(x)) of any continuous function `f` on the non-negative real numbers is again continuous (because `abs` is).
One approach to avoiding the whole question of abs(x) would be to use cosine instead of sine, since cosine is symmetric across x = 0.
I don't think that this would work, since the division by an odd function would just make it odd again (that is, `sin(x)/x` is even precisely because `sin` is odd). Note, however, that blaming `abs` for discontinuity (or non-differentiability) is a bit of a red herring; any even function on the real line, no matter how smooth, can be written as a composition with `abs`.
My point was simply that the "spike" the article goes on about (which I earlier called a discontinuity, oops) shows up because the author is reflecting his plot around the Y axis. The article makes it sound like the spike was some physically significant feature of the function being graphed.
My point was more that there's nothing inherently 'spiky' about even functions; as your own example of cosine shows, they can be just as smooth (or spiky) as any other function.