Let y( x) be a real-valued function defined on the interval 0 ≤ x ≤ 1 by means of the equations
( a) Show that the equation
represents an are C that intersects the real axis at the points z = 1 /n( n = 1,2,...) and z = 0, as shown in Fig. 38.
( b) Verify that the arc C in part( a) is, in fact, a smooth are.
Suggestion: To establish the continuity of y( x) at x = 0, observe that
When x > 0. A similar remark applies in finding yʹ(0) and showing that yʹ( x) is continuous at x = 0.