Studying Mathematics Thread

[polymath257]

Say we have a function defined as follows:

f(x) = x^2 when x < 1
     = 2x-1 when x >= 1

Is f(x) differentiable at point where x = 1?

One would think at first glance the answer is "no" because the two parts of the function are two different functions. However, the limit of the slope of the tangent line as it approaches the point of interest (where x = 1) from the left is the same as the limit of the slope as it approaches the same point of interest from the right. Using the power rule, we get the same thing. The derivative at point where x = 1 is consistently 2.

This means that f(x) is actually differentiable at point where x = 1. Did I get this correct? Or am I missing something here?

That is correct. if you do the *definition* of the derivative in terms of a limit for this function, you use the x^2 for the left limit and 2x-1 for the right limit. Since both limits in the definition are 2, f'(1)=2 and the function is differentiable there. It is even continuously differentiable (the derivative is a continuous function). But, it is not differentiable two times: the derivative of the derivative is not defined at x=1.