Baby Rudin exercise 8.26 Solution Manual: 
https://minds.wisconsin.edu/bitstream/handle/1793/67009/rudin%20ch%208.pdf?sequence=4&isAllowed=y
In the second part of the exercise, we are asked to prove exercises 24 and 25 without the assumption of differentiability of $\gamma(t)$. However, isn't the definition of $Ind(\gamma)$ that was defined in exercise 23 contingent on the differentiability of $\gamma(t)$?
Could anyone provide some insight regarding this?

Read the set up to the exercise a little more carefully. We formulate the winding number of a continuous curve by first approximating the continuous curve with a trigonometric polynomial (which is continuously differentiable) and then taking the winding number of that. 
