I have been thinking how "smoothness" would have got a Mathematically rigorous definition. For the same, if we look at our intuition, "smoothness" of a function (a real valued one variable function) would mean that we should not have any "sharp" points in its graph in $\mathbb{R}^2$.
This leads me to think "What do we mean by sharp points?" Intuition tells that a point in the graph will be "sharp" if it "suddenly changes", as we change our input variable. Mathematically, this would mean that if we approach this point from, say "left", then we have a different "slope" and if we approach from right, we would have a different slope. Hence, at a "sharp" point, at least intuitively, the function would not be differentiable.
Coming to the smoothness, intuition tells us that if a function has no sharp point, then it must be smooth. Therefore, if the first derivative exists (and is continuous), then the function is smooth. However, all sources tell that a function is smooth if it has derivatives of all orders.
My question is why higher order derivatives are required? What is the problem if we consider only the first derivative?