## Derivative of a function on $C([0,1], \Mathbb{R}$ [closed]

Hi, I'm stuck in the following problem: I have considered, for h>0:

I have considered, for h>0:

$\frac{\varphi(f(x)+t g(x))-\varphi f(x)}{t}$

While on the other way: $\frac{\varphi(f(x)+t g(x))-\varphi f(x)}{t}\ge$ But now? These inequalities don't give me information... $\frac{\varphi(f(x)+t g(x))-\varphi f(x)}{t} \le$ . Where $f$ and $g \in C^0 ([a,b],\mathbb{R})$.

I have considered, for h>0:

$\frac{\varphi(f(x)+t g(x))-\varphi f(x)}{t}$

While on the other way: $\frac{\varphi(f(x)+t g(x))-\varphi f(x)}{t}\ge$ But now? These inequalities don't give me information...

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 Welcome to MO. It is good if you let us know that the problem is solved, however it is neither common here nor helpful to remove the question via an edit in the process. (It might be you can truly delete it, look for text saying deleted below the tag, not sure though this works; this would be fine, but is not necessary.) – quid Oct 20 at 14:38 Did the problem resurface, or do you see any other problem with displaying both your version together? Well, it is not that big a deal either way. I will leave this alone for now. – quid Oct 20 at 14:41