# Is this problem right?

Let $$f$$ be a continuous function from an open interval $$I\subseteq\mathbb{R}$$ to a Banach space $$E$$. The problem/exercise asks to prove that $$f$$ is differentiable in $$x_0\in I$$ if and only if the limit $$\lim_{(h,k)\to (0,0)}\frac{f(x_0+h)-f(x_0-h)}{h+k}$$ exists when $$h,k\gt 0$$.

Shouldn't it be $$\frac{f(x_0+h)-f(x_0-k)}{h+k}$$ otherwise i don't see what is $$k$$ doing there in the denominator. I've been able to do the exercise asuming it wanted to say "$$k$$" there because, if not, then $$0=\lim_{k\to0^+}(\lim_{h\to0^+})$$ can be very different to $$\lim_{h\to0^+}(\lim_{k\to0^+})$$ and they should be equal. Should I try it as it is?? The book this is from doesn't mess around and the $$h$$ could be on purpose to make the exercise very difficult.

Thanks

• Surely, there is a typo in the exercise. – Kabo Murphy Jan 3 at 23:24

## 1 Answer

You are right. It is obviously a typo.