I came to Illinois Tech as an Instructor in January of 1957 having just received my PhD from R. L. Moore. I knew of and had a great deal of respect for Professor Menger and was anxious to meet him. Almost immediately after I arrived in the department I was told that he was quite irritable and that whatever else I did I was not to criticize his book on Calculus.
I did meet Professor Menger very shortly after my arrival and found him outgoing and friendly contrary to what I had been told. We met frequently in his office to discuss mathematics. One day he was talking at his usual breakneck speed and mentioned with his accent what sounded like a "nowerdiferentablfunction." After asking him to repeat it several times I finally understood that he was saying a nowhere differentiable function and I asked him what that meant. He explained and wanted to show me an example. I declined saying I would like to work on the problem. The next day he patiently sat and listened to me give, as best I recall, a horribly complicated example of an arc in the plane and a parameterization of it so that the parameterizing functions were nowhere differentiable. I am sure my explanation took over an hour. After my discourse he asked if he could show me a simple example and he did so in a very few minutes.
We became good friends and he called my apartment frequently to ask questions or discuss mathematics. He raised one question which I settled and he tried to get me to submit it to Fundamentae Mathematicae but I declined saying I thought it was too simple. As summer approached he offered to pay me from a grant he had if I would write up my proof and submit it. I accepted his offer. which allowed me the freedom to do research in the summer. The solution of that problem became my first publication. Recently I came across a copy of that paper and I had considerable difficulty understanding what I had done.
I recall one instance when I was waiting outside a class he was teaching to talk to him. I watched him lecturing at a rapid pace, filling the board over and over again. He continued until the bell rang and then some and no one moved to leave. Finally he left the room and the students followed in pairs or groups talking animatedly about what they had just heard. As a student of R. L. Moore I almost never lecture but this convinced me that Menger got his students just as excited about mathematics by lecturing as Moore did using his methods.
Finally I will say that I was a severe critic of his book. I claimed that what he had done was to introduce notation that allowed students to do the same mindless manipulation they did with the standard notation but be mathematically precise. He claimed it was important that the students be precise in what they did whether they understood it well or not. We spent many pleasant hours arguing these points. I understand better now what he meant.
I treasure my memories of Professor Karl Menger.