I want to fadein a UI image from transparent(alpha=0) to alpha=1, i thought my approach should be right, but it doesn't work, the image is not changing.

This is the code that i tried for doing that:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;
public class Fadein : MonoBehaviour {


    public float FadeRate;
    private Image image;
    private float targetAlpha;

    // Use this for initialization
    void Start()
    {
        image = GetComponent<Image>();
        Material instantiatedMaterial = Instantiate<Material>(image.material);
        image.material = instantiatedMaterial;
        targetAlpha = image.material.color.a;

        Invoke("startFadein", 1);

    }

    IEnumerator FadeIn()
    {
        targetAlpha = 1.0f;
        Color curColor = image.material.color;
        while (Mathf.Abs(curColor.a - targetAlpha) > 0.0001f)
        {
            curColor.a = Mathf.Lerp(curColor.a, targetAlpha, FadeRate * Time.deltaTime);
            image.material.color = curColor;
            yield return null;
        }
    }

    void startFadein()
    {

        StartCoroutine(FadeIn());
    }
}

The image is not changing. But i tried fadeout by using this code, from 1 to 0, it just worked, i have no idea why the fadein doesn't work?

`image.material.color` is not what you think it is

With a few debug lines I was able to determine that the image material's alpha reports that it is 1 even when I set the image color multiplier to 0.

If I override the curColor to have a 0 and let the loop do its thing, the image never appears either.

This is because this:

Is not image.material.color. It's image.color.

Thus your fixed code would be:

IEnumerator FadeIn() {
    targetAlpha = 1.0f;
    Color curColor = image.color;
    while(Mathf.Abs(curColor.a - targetAlpha) > 0.0001f) {
        Debug.Log(image.material.color.a);
        curColor.a = Mathf.Lerp(curColor.a, targetAlpha, FadeRate * Time.deltaTime);
        image.color = curColor;
        yield return null;
    }
}

Additionally a few other things:

Your code does not lerp the color linearly. I'm sure you knew that, and you're probably fine with that, but I figured I'd point it out.
You don't need Invoke("startFadein", 1);. You can just call StartCoroutine(FadeIn()); and put yield return new WaitForSeconds(1) at the top.
Your image will never actually reach the target value, it'll be close, but not equal. You can fix this by putting curColor.a = targetAlpha; image.color = curColor; after the while loop.