# Differentiability and Continuity

If

*f*(*x*) is differentiable at every point of its domain, then it must be continuous in that domain.

*Notes:*- The converse of it is false; that is, there are functions that continuous but not differentiable. For instance, the function
*f*(*x*) = |*x*| is continuous at 0 because

- The converse of the above result is not true, i.e., if
*f*(*x*) is continuous at*x*=*a*then it may or may not be differentiable at*x*=*a*. - If
*f*(*x*) is differentiable then its graph must be smooth, i.e., there should be no break or corner.*f*(*x*):- Differentiable ⇒ Continuous
- Continuous ⇒ May or may not be differentiable
- Not continuous ⇒ Not differentiable

# How can a function fail to be differentiable?

The function

*f*(*x*) is said to be non differentiable at*x*=*a,*if- both
*Rf*′(*a*) and*Lf*′(*a*) exist but are not equal - either or both
*Rf*′(*a*) and*Lf*′(*a*) are not finite - either or both
*Rf*′(*a*) and*Lf*′(*a*) do not exist

The function

*y*= |*x*| is not differentiable at 0 as its graph changes direction abruptly when*x*= 0. In general, if the graph of a function has a “corner” or “kink” in it, then the graph of*f*has no tangent at this point and*f*is not differentiable there. [In trying to compute*f*′(*a*), we find that the left and right limits are different.]If

*f*is not continuous at*a*then*f*is not differentiable at*a*. So, at any discontinuity (for instance, a jump discontinuity)*f*fails to be differentiable.A third possibility is that the curve has a vertical tangent line when

*x*=*a*, that is,*f*is continuous at*a*and .This means that the tangent lines become steeper and steeper as

*x*→*a*. Following figure illustrates the three possibilities that we have discussed.