Question

Can a point of discontinuity have a tangent line?

Answer 1

No. Never, in fact, unless you live in some strange universe where mathematics itself ceases to exist.

Explanation:

When we talk about tangent lines, we usually assume that the point we're finding the tangent line of is continuous. If it's not, `f(x)` does not exist on that point and neither does `f'(x)` (if you haven't learned derivatives yet, it's fine - just know for now that `f'(x)` is the slope of the tangent line). For example, because `f(x)=1/x` is not continuous at `x=0`, the tangent line to `f(x)` does not exist at `x=0`. It would essentially be a straight line, but because the slope of a straight line is undefined, so would the line itself.

Of course, there's a strict mathematical definition/proof to show that discontinuities don't have tangent lines, but I'm not a big fan of those things - they tend to interfere with your understanding of concepts, especially at the introductory level. So for now, all you need to know is that because `f(x)` is not defined at discontinuities, neither is the tangent line at `x`. You simply can't draw tangent lines at these points.