Worksheet for Section 2.6

Secton 2.5 is about the connection between derivatives and continuity. First, a function $f(x)$ is differentiable at $c$ if the following limit exists:

$$\lim_{h\to 0}\frac{f(c+h)-f(c)}{h}$$

(In other words, $f(x)$ is differentiable at $c$ if the derivative of $f(x)$ at $c$ exists.) This can be extended similar to how the definition of continuity was extended, to talking about differentiability on an interval: $f(x)$ is differentiable on an interval $I$ if it is differentiable at $c$ for all $c$ in $I$. There are several ways in which a function can fail to be differentiable at a point $x=c$:

• the function is not continuous at $x=c$ -- more on this below
• the graph of the function has a sharp corner at $x=c$ -- for example, consider the derivative of $f(x)=\vert x\vert$ at $x=0$
• the graph has a vertical tangent line at $x=c$ -- for example, consider the derivative of $g(x)=\sqrt[3]{x}$ at $x=0$

The first possibility above suggests a connection between differentiability and continuity. Since both are defined as limits, the connection is quite close: if $f(x)$ is differentiable at $c$), then $f(x)$ is continuous at $c$. However, this does not work in reverse -- there are lots of functions which are continuous but not necessarily differentiable at a point. The easiest examples are $f(x)=\vert x\vert$ and $g(x)=\sqrt[3]{x}$ above -- in both cases, determine first if the functions are continuous at $x=0$, then try to compute their derivatives at $x=0$.