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I want to apply a sticker to a glass bottle that has transparent areas. I got it to work that the right parts are see-through. My problem is that i have white blurry areas where the sticker transitions from not transparent to transparent. But I want sharp edges around the border of the colored part. In other words:

I want the texture to be either 100% transparent or 0%. Nothing in between.

Here is a screenshot of the texture, my node setup and my current render result. Screenshot

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You were on the right track: you can use a math node in between the alpha output of your texture and the mix shader. set it to "less than" and input a number between 0 and 1. if you input 1, only 100% opaque pixels will be diffuse, all other pixels are glass.

Here is the node setup (I used transparent and emission shaders for demonstration purposes. this works with your node setup as well), the texture I used and the result:

Node Setup

Image Texture

result

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    $\begingroup$ As an alternative to the math node you can use a color ramp node. $\endgroup$ – cegaton Feb 26 '18 at 14:01

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