Note: A local homeomorphism is still available if one assumes everywhere differentiability of the function on the domain although the result is a bit complicated. For reference visit terry tao’s blog which has a good explanation on it.
A function is C^k if its k-th order derivative exists and is continuous. Thus, C^1 functions are continuously differentiable and C^2 functions are twice continuously differentiable. For example, f(x)=x^{4/3} is C1 but not C2 on R, since its second-order derivative is discontinuous at x=0.
Note: A local homeomorphism is still available if one assumes everywhere differentiability of the function on the domain although the result is a bit complicated. For reference visit terry tao’s blog which has a good explanation on it.
Sir, what is the difference between C1 and C2 function?
A function is C^k if its k-th order derivative exists and is continuous. Thus, C^1 functions are continuously differentiable and C^2 functions are twice continuously differentiable. For example, f(x)=x^{4/3} is C1 but not C2 on R, since its second-order derivative is discontinuous at x=0.