Where is f(x) differentiable?
The graph above is f(x).
I initially thought that it would be differentiable for all values of x between -2 and 4 since it seems to be defined for all those values. However, that answer was wrong.
Any help?
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$\begingroup$Differentiable is not equivalent to defined for all values. The real definition of differentiable is that the derivative of the function exists at all points (on the interval). This means that since $f'(-1)$ is undefined ($\lim_{x\rightarrow -1^-}f'(x)$ is clearly much greater than $\lim_{x\rightarrow -1^+}f'(x))$, the function is not differentiable on the domain given.
$\endgroup$ $\begingroup$The function is differentiable when $$\lim_{x\to\ a^-} \frac{dy}{dx} = \lim_{x\to\ a^+} \frac{dy}{dx}$$ Unless the domain is restricted, and hence at the extremes of the domain the only way to test differentiability is by using a one-sided limit and evaluating to see if the limit produces a finite value.
Hence, the function described by the graph provided is differentiable on the interval $$x \in [-2,-1) \cup (-1,0) \cup (0,1) \cup (1,3) \cup (3,4]$$
$\endgroup$ 1 $\begingroup$To be differentials, a function must be continuous and have no sharp corners. (Endpoints of the function are considered sharp corners.)
Looking at the graph, the function is discontinuous at
- $x<-3$, because the function is not defined
- $x=-1$, because the point there is removed from the graph to either side
- $x=0$, because of an asymptote
- $x=1$, because the function jumps
- $x>4$, because the function is not defined
The function has sharp corners at
- $x=-2$, because the curve has an endpoint there
- $x=-1$, because the ‘slope’* coming in from the left is different than coming in from the right
- $x=3$, because the ‘slope’* coming in from the left is different than coming in from the right
- $x=4$, because the curve has an endpoint there
Everywhere else, the function is differentiable.
Remember—differentiable is a subcategory of continuous.
* i.e., of the tangent line
PS: You know a function is continuous if you can draw its graph without picking up your pencil.
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