Thread: Find where f is continuous and differentiable

For all n in N (natural numbers), let f(x):= x^n if x >= (gr/eq to) 0 and f(x)=0 for x<0. For which values of n is f' continuous at 0? Which values of n is f' differentiable at 0? I can understand why this is, but I don't know how to prove it using limit definitions. Thanks for your help

Originally Posted by mremwo

For all n in N (natural numbers), let f(x):= x^n if x >= (gr/eq to) 0 and f(x)=0 for x<0. For which values of n is f' continuous at 0? Which values of n is f' differentiable at 0? I can understand why this is, but I don't know how to prove it using limit definitions. Thanks for your help

You need to check the limit from both sides

The left one is easy. let

Then

Now the right limit is where this get interesting!

now we need to analyze



This need to equal the limit form the left! So what must the values of be?