2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Learning objectives

—FUN-2.A—Explain the relationship between differentiability and continuity• FUN-2.A Explain the relationship between differentiability and continuity.• FUN-2.A.1 If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of f, then it is not in the domain of f '.• FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.• Enduring understanding FUN-2: Recognizing that a function’s derivative may also be a function allows us to develop knowledge about the related behaviors of both.

If $f$ is differentiable at $x=a$ , then $f$ is continuous at $x=a$ . Differentiability is therefore the stronger condition. The converse is false: a function can be continuous at a point without having a derivative there.

When $x\ne a$ , write $f(x)-f(a)=\frac{f(x)-f(a)}{x-a}(x-a).$ If $f'(a)$ exists as a finite two-sided limit, the first factor approaches $f'(a)$ while the second approaches $0$ . Their product approaches $0$ , so $f(x)\to f(a)$ : this is continuity at $a$ .

For $f(x)=|x|$ at $x=0$ , the graph is continuous, but the difference quotient approaches $-1$ from the left and $1$ from the right. The two-sided derivative does not exist because those limits disagree.

For $f(x)=\sqrt[3]{x}$ at $x=0$ , the graph is continuous, but its tangent is vertical. The difference quotient grows without a finite limit, so $f'(0)$ does not exist as a finite derivative.

Use the implication in the correct direction: differentiable $\Rightarrow$ continuous. A discontinuity guarantees non-differentiability, but continuity alone never guarantees differentiability. Also, if $a$ is not in the domain of $f$ , neither continuity nor differentiability at $a$ is defined.

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