Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Differentiability.


 

As we have emphasised on various pages and videos in this website, differentiation is all about gradients. So differentiability is all about determining whether or not we can calculate the gradient of a given curve at all points along the curve.

 

Discontinuity and differentiability

If a curve is continuous at a point, that means that we can trace the curve with our finger and not lift our finger off the paper.

If however we do lift our finger, then the curve is discontinuous and so we cannot determine its gradient at that point.

A curve cannot be differentiable at a point of discontinuity although it could be differentiable at all other points depending on its definition.

 

Testing for differentiability

In an concept analogous to that of continuity, a curve is said to be differentiable if the gradient at a given point:

Remember that a derivative measures the rate of change at a given point (or at a given time or place).
So to conclude that a curve is differentiable is really saying that the rate of change is smooth and continuous at that point.

Most curves which are continuous at a given point are also differentiable at that point. The gradient is therefore the same on either side or the values are changing smoothly.

For example, the graph of shown to the right has:

  • no breaks - so the curve is continuous;
  • at any point, the slope of the line is the same whether we approach the selected point from above or from below.

For this graph of , the description is the same:

  • no breaks - so the curve is continuous;
  • at any point, the slope of the line at the selected point becomes the same gradient as we approach the selected point from above or from below.

 

For the graph , the curve is discontinuous at x = 0. It therefore is not differentiable at x = 0.

If the domain is restricted to one side of the
y-axis, then within that restricted domain, the curve is both continuous and differentiable.

The function y = |3x| is continuous.

The gradient for x < 0 is negative and the gradient for x > 0 is positive.

The gradient at x = 0 cannot be determined and does not follow from the lines on either side. Hence the function is not differentiable at x = 0.

 

No formal proof of differentiability is needed but you need to be able: