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?
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?
