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.


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    $\begingroup$ Surely, there is a typo in the exercise. $\endgroup$ – Kabo Murphy Jan 3 at 23:24

You are right. It is obviously a typo.


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