number of tangents

Question

Find number of points where tangents does not exist for the curve y =sgn(x² -1 )

Answer 1

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Answer 2

This function is a step function with discontinuities at x = -1 and x = 1.   At those points the function is not defined, and even more importantly, the limits of the function on the left and right at each of those points of discontinuity are different so the limit of f as x approaches either -1 and 1 are non existent.

 

The above is sufficient to show that the derivative does not exist at that point.  Accordingly a tangent to the curve, which would have a slope given by the derivative does not exist for those two points.   At all other points, the tangent exists and is horizontal.