the reflection map z-bar
The reflection map is f(z) = z-bar, the complex conjugate: if z = x + i y then z-bar = x - i y. Geometrically it flips the plane across the real axis like a mirror — every point is sent to its mirror image below (or above) the horizontal axis. Points already on the real axis stay put; the point i goes to -i; the point 2 + 3i goes to 2 - 3i.
It preserves distances (|z-bar| = |z|, since conjugation does not change the modulus) and it preserves the size of angles, so it looks almost like a rotation or translation. But there is a crucial difference: it reverses the sense of angles. If you sweep counter-clockwise from one curve to another by 30 degrees in the z-plane, in the image you sweep clockwise by 30 degrees. A map that preserves angle sizes but flips their orientation is called anti-conformal, the mirror-image cousin of a conformal map.
This orientation-reversal is why z-bar matters as a cautionary example. It is perfectly continuous and perfectly distance-preserving, yet it is not complex-differentiable anywhere: writing f = u + i v gives u = x, v = -y, which fails the Cauchy-Riemann equations. So z-bar is the cleanest illustration that 'smooth and nice as a real map' is far weaker than 'holomorphic'. Its building-block role also appears in the Schwarz reflection principle, which extends functions across the real axis using conjugation.
Under f(z) = z-bar, the triangle with vertices 0, 1, i (which you traverse counter-clockwise) maps to the triangle 0, 1, -i — same shape and size, but now traversed clockwise. The flip across the real axis is visible at a glance.
Conjugation is a mirror flip: it keeps lengths and angle sizes but reverses orientation.
z-bar is the standard counterexample to 'continuous implies differentiable'. It is as smooth as any real map of the plane, but because it reverses orientation it satisfies none of the Cauchy-Riemann conditions and is nowhere holomorphic.