Conditions for differentiability for a single variable

For a function to be differentiable it has to be continuous. However, being continuous does not imply in differentiability. The graphical way to explain this is to show a function that is continuous but not smooth. The easiest example is [math]\displaystyle{ f(x) = |x| }[/math]. At the origin the function is continuous because both sided limits converge to zero. But the tangent line there cannot be defined because if we try to use the tangent line idea we have a problem: a division by zero occurs there. We didn't say that the previously mentioned function is not differentiable though. At the origin it isn't, but everywhere else it is differentiable. To put it in another words: every derivative is a limit that does exist; but not every limit is a derivative. If the limit doesn't exist at a point, then the derivative doesn't exist too at the same point. On the other hand, the existence of a limit at a point doesn't guarantee that the derivative also exists there. This is caused by the derivative being defined as a special limit.

A natural question: we learn that a function can be discontinuous everywhere. Is there a function that is continuous everywhere, yet non differentiable everywhere? Yes, they exist and one example is the Weierstrass function. The function is continuous because if we take two points of it, there are never gaps in between them. The problem with differentiating it is that, although there aren't gaps in between two consecutive points, we can never see constant variations in between a point and the next point. To simplify the concept let's say that the function is defined at a point [math]\displaystyle{ a }[/math]. Then for a 0.001 increment we have that [math]\displaystyle{ f(a + 0.001) }[/math] and [math]\displaystyle{ f(a - 0.001) }[/math] are almost the same value, with a distance that is close to zero. Weierstrass defined a function such that no matter if the distance between two consecutive points is small, the rate of change between [math]\displaystyle{ f(a + 0.001) }[/math] and [math]\displaystyle{ f(a - 0.001) }[/math] is significant to the point of making this function non-smooth everywhere. Another perspective: every process has some high frequency variations that we disregard when we model them with the common functions that everybody knows from school. The Weierstrass function is such that the high frequency variations are part of it and cannot be "smoothed out" in any way. We just can't "iron" this function as we do with clothes.