Is $ \left| \lim_{t \rightarrow 0}{\frac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \frac{|f(x+ty) - f(x)|}{|t|}}$

Question

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true for all $x,y \in X$ that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \dfrac{|f(x+ty) - f(x)|}{|t|}}?$$

I think it is true when $X = \mathbb{R}^n$, but I don't know how to show it.

Answer 1

Fixing $x, y\in X$ and consider $$g(t) = \frac{f(x+ ty) - f(x)}{t}.$$ Then your question reduces to an easy question in real analysis. You need only that $|\cdot|$ is a continuous function.

Answer 2

Actually equality holds because the norm $|\cdot |$ is a continuous function. To show that it is continuous, simply note that it is Lipschitz since for all $x,y$ $$||x|-|y|| \le |x-y|$$