f(x)=|x| is an example of a function that is graphed as two semi-infinite straight lines radiating from the origin. When x<0 f(x)=-x and when x>0 f(x)=x. These two lines meet one another at right angles at the origin forming a wide V shape. The origin is a definable point (0,0) but the slope at this point is undefinable. Therefore the function is continuous but not differentiable over the whole domain.
All functions of the form f(x)=|mx+c| where m and c are constants, are similarly continuous over the domain but when x=-c/m the function cannot be differentiated. This can also be seen by finding the derivative of f(x), which is m(mx+c)/|mx+c|. When x<-c/m, mx+c<0 and the derivative is -m and when x>-c/m the derivative is m. When x=-c/m the derivative is not defined. (f(x)=|x| is the special case when m=1 and c=0.)
Another function which is continuous but is totally undifferentiable is of the type x=a where a is a constant, because dy/dx is effectively infinite at every point.