Why do we see our image on the concave mirror when it it actually a real image and formed in front of the mirror


If the image is not upside-down, then it is not the image. That is, when you are too close to the mirror, you can't see the image because it is formed behind your head. Instead, the lens in your eye focuses rays that are already converging, creating the illusion that the object is large and beyond the mirror.

When the image appears upside-down and small, then you are seeing the image as if it were an object, and it appears to be in front of the mirror.

By the way, I have always been fascinated with the transition as I back away from a concave mirror. The image of my face goes from rightside-up to upside-down without performing a flip or a rotation. Instead it becomes increasingly de-focused until every point of the image is an average of the the colors of all the points in the scene. That occurs in the so-called Fourier transform plane. Increasing the distance further, the image becomes less blurred, but is now smaller and upside-down.
Real image means that if you put a screen at the image oint you will see the object projected on the screen.
However, if you do not use a screen but look at the real image, you actually see the light rays coming from the mirror and you assume that the image is behind the mirror.
This also applies in the plane mirror; we perceive the image to be behind the mirror since we get the light rays reflected/coming from the mirror.

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