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If we consider an object with a mirror finish (i.e an object with very little micro-imperfections, i.e an object with very low roughness, i.e. a very smooth object), the object will act as a mirror. And in a mirror we can see sharp specular reflections of the environment (including objects other than the light source).

I continue quoting from Wikipedia:

A piece of highly polished white marble remains white; no amount of polishing will turn it into a mirror. Polishing produces some specular reflection, but the remaining light continues to be diffusely reflected.

However, even mirrors have some diffuse reflection. It just happens to be very, very, hard to notice. But we can make it more evident by making repeated reflections, by reflecting a mirror in a mirror.

Picture taken from the video What Color is a Mirror by VSause.

We observe that the repeated reflections are less bright, and have a tint. Usually a green tint, due to the chemistry of the glass.

By the way, the the video includes a picture that suggest diffuse reflection happens due to roughness. So, I'll reiterate that we attempt to model scattering as diffuse reflection.

You get specular and diffuse reflections on every point.

The specular reflection is negligible everywhere except on the specular highlight. I mean, the specular reflection decays away from the specular highlight until it is virtually zero.

While the way specular reflection decays attempts to model roughness. The diffuse reflection attempts to model scattering.

That is, the definition of diffuse reflection holds. But it does not account for how its contributions is combined with the others.

Every point with incident light has both specular reflections and diffuse reflections. There are no portions of the object that are getting specular reflection but no diffuse reflection, nor viceversa. There are no portions of the object that are inherently specular or diffuse. In fact, specular reflection and diffuse reflection happen together.

Instead, for a single point on the surface of the object with incident light:

• Specular reflection dominates over the diffuse reflection.
• Diffuse reflection dominates over the specular reflection.
• Or somewhere in the middle. Because the decay of the specular highlight is not abrut.

The specular reflection is negligible everywhere except on the specular highlight. I mean, the specular reflection decays away from the specular highlight until it is virtually zero.

I want to point out that "specular highlight" and "specular reflection" are not the same thing. In the prior answer, on first paragraph, I took the care to define "specular highlight":

The specular highlight (…) is an specular reflection of the light source.

Emphasis added on "of the light source".

Other objects aside from the light source also appear in the specular reflection. But only the reflections of the light source are specular highlights.

While the way specular reflection decays attempts to model roughness. The diffuse reflection attempts to model scattering.

That is, the definition of diffuse reflection holds. But it does not account for how its contributions is combined with the others. For any lighting model, the contributions of specular reflection, diffuse reflection, (and any other component) must be combined to make up the final shading of the pixel. This is usually a weighted sum.

The final shading of every pixel is a combination of the contributions of the specular reflection, diffuse reflection (and ambient and any other components according to the lighting model).

But away from the specular highlight, the specular contribution is negligible. Leaving only the diffuse and ambient reflections noticiable.

On the other hand, on the specular highlight, the specular reflection is very intense, and as a consequence we do not notice the contributions of diffuse and ambient reflection.

The specular reflection, diffuse reflection (and ambient and any other components according to the lighting model) contribute to the final shading of every pixel.

But away from the specular highlight, the contribution of the specular reflection is negligible. Leaving only the contribution of the diffuse reflection (and ambient reflections or any other component according to the lighting model) noticiable.

On the other hand, on the specular highlight, the specular reflection is very intense. As a consequence we do not notice the contributions of the diffuse reflection (and ambient reflection and so on).