Please give an example of a continuous function $f:[0,1]\to\mathbb R$ which doesn't have a derivative at any point. I can't think of anything, can someone help please?
There is a well-known example of a continuous function that is not differentiable by Weierstrass:
Just take any function which is nowhere continuous.
Finding an $f$ which is continuous but nowhere differentiable is much harder. The most famous example is the Weierstraß function, which was explicitely constructed to have that property.
Before its construction, it was conjectured, that every continuous function was differentiable except on a set of isolated points.
So its not surprising that you cannot think of anything