Differentiation · Definition

Differentiability and continuity

real analysisdifferentiation

Before you start you need to know

It ranks two very common pointwise properties of a function: being differentiable demands more than being continuous.

Definition

If a function \(f\) is differentiable at a point \(c\), then \(f\) is continuous at \(c\). The converse is false.

\[f'(c)\text{ exists}\ \Longrightarrow\ \lim_{x\to c}f(x)=f(c)\]

The proof fits in two lines. For \(x\ne c\) write \(f(x)-f(c)=\frac{f(x)-f(c)}{x-c}\,(x-c)\). As \(x\to c\) the first factor tends to \(f'(c)\) and the second to \(0\), so the product tends to \(0\) and \(f(x)\to f(c)\). The idea is that a difference quotient with a finite limit forces the numerator to vanish.

The converse fails in several ways. \(|x|\) has a corner at \(0\): it is continuous, with one-sided slopes \(\pm1\). \(x^{1/3}\) has a vertical tangent. \(x\sin(1/x)\), with value \(0\) at the origin, has a quotient \(\sin(1/h)\) that oscillates with no limit. In all three cases continuity survives but the derivative does not exist.

It helps to read the theorem in its contrapositive form: if \(f\) is not continuous at \(c\), it cannot be differentiable there. A function with a jump, such as the one that is \(0\) to the left and \(1\) from the origin on, is ruled out at once, without computing any limit.

There are functions continuous on the whole line that are differentiable nowhere (Weierstrass's classical example), so the gap between the two properties can be enormous. In the next concepts, Rolle's theorem and the mean value theorem, continuity on the closed interval and differentiability on the open one are requested separately precisely because of this difference.

Think about it

If \(f\) and \(g\) are continuous at \(c\) and \(f+g\) is differentiable at \(c\), must \(f\) and \(g\) be?

No. For instance \(f(x)=|x|\) and \(g(x)=-|x|\) are continuous, not differentiable at \(0\), and their sum is \(0\), differentiable everywhere.

Can a function differentiable at \(c\) be discontinuous at points arbitrarily close to \(c\)?

Yes. Being differentiable at a single point only requires continuity at that point: the Dirichlet function times \(x^2\) is differentiable at \(0\) and discontinuous at every other point.

Exercises 10

  1. Which statement is true for a function \(f\) and a point \(c\)?
  2. What happens to \(f(x)=|x|\) at \(c=0\)?
  3. Let \(f(x)=x^{1/3}\). What can be said at \(c=0\)?
  4. Let \(f(x)=x^2\) if \(x\le1\) and \(f(x)=ax+b\) if \(x>1\). If \(f\) is differentiable at \(1\), what is \(a+b\)?
  5. Let \(f(x)=x^2\sin(1/x)\) for \(x\ne0\) and \(f(0)=0\). What is \(f'(0)\)?
  6. If \(f\) is not continuous at \(c\), what follows?
  7. In the proof that differentiable implies continuous, which identity is used for \(x\ne c\)?
  8. Let \(f(x)=1\) if \(x\ge0\) and \(f(x)=0\) if \(x<0\). Is \(f\) differentiable at \(0\)?
  9. Let \(f(x)=x\sin(1/x)\) for \(x\ne0\) and \(f(0)=0\). Which statement is true at \(0\)?
  10. At how many points is \(f(x)=|x|+|x-2|+|x+3|\) not differentiable?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §5.2, Teorema 5.2.3
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 5, Teorema 5.2
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §6.1, Teorema 6.1.2