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Can someone give me an example of a nontrivial endomorphism of the space of continuous functions $R$ to $R$, besides derivation and multiplication by a scalar?

I know other endomorphisms exists, but I don't want some pathological thing I would like a concrete, explicit example besides the ones I already know.

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    You can multiply by any fixed continuous function, e.g. the map that sends $f$ to $g$ where $g(x)=(\sin x) f(x)$. You can integrate, e.g. the map that sends $f$ to $g$ where $g(x)=\int_0^x f(t)\,dt$. You can translate, e.g. sending $f(x)$ to $f(x-1)$. – 2017-02-18
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    Hmm why didn't I think of that? – 2017-02-18

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Is such example $$\Phi(f)(x)=f(2x)$$ good for you? This is rescalling in the argument.

This example has a quite natural extension. Let $h:\Bbb R\to\Bbb R$ be a continuous bijection. Define $$\Psi(f)(x)=f\bigl(h(x)\bigr).$$

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Any homeomorphism $\tau:\mathbb{R}\to\mathbb{R}$ induces an algebra automorphism $\Phi_\tau:C(\mathbb R)\to C(\mathbb R)$ defined by $$\Phi_\tau(f)(x)=f(\tau(x))\qquad (x\in\mathbb{R}).$$

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    Here translations in the argument could be important. For instance, invariant means are based on them. – 2017-02-18