Differentiability#
Definition#
Definition
The function \(f(x)\) is differentiable at \(x = a\) if the following limit exists.
Intuitively, \(f(x)\) is differentiable at \(x=a\) if we can zoom in on the graph of \(y=f(x)\) at the point \((a,f(a))\) until the graph looks like a single uninterrupted straight line.
Differentiability versus Continuity
If a function is differentiable at \(x=a\), then it is continuous at \(x=a\). This also means that if a function is not continuous at \(x=a\), then it is automatically not differentiable at \(x=a\).
However, if a function is continuous at \(x=a\), then it is not necessarily differentiable at \(x=a\).
Example 1#
Continuous and differentiable
The function defined by the following graph is continuous and differentiable at every point on the graph.
Intuitively, the graph can be drawn without picking up a pencil from the paper (i.e., it’s continuous) and we can zoom in close enough to any point on the graph so that the graph looks like a straight line (i.e., it’s differentiable).
Long Text Description
The graph of a continuous function that increases before x = -1, decreases between x=-1 and x=1, and then increases after x=1. The graph does not have any sharp corners or vertical tangent lines.
Example 2#
Continuous but not differentiable at a point
The functions defined by the following two graphs are continuous at every point on the graph. However, both functions are not differentiable at \(x=1\).
For the graph on the left, the function is not differentiable at \(x=1\) since there is a sharp corner on the graph at \(x=1\) (i.e., \(f'(1)\) does not exist). The function is differentiable everywhere else.
For the graph on the right, the function is not differentiable at \(x=1\) since it has a vertical tangent line at \(x=1\) and vertical lines do not have a slope associated with them (i.e., \(f'(1)\) does not exist). The function is differentiable everywhere else.
Long Text Description
There are two graphs displayed. The first graph corresponds to a function that increases linearly with a slope of postive 1 to the point (1,3) where it reaches a corner and then decreases linearly with a slope of -1.
The second graph corresponds to a function that increases from left to right. The curve becomes increaseingly steep as x approaches 1 from the left and ultimately becomes vertical at x = 1. The curve continues to increase for x greater than 1 but flattens out as x increases.
Example 3#
Not continuous and not differentiable at a point
The functions defined by the following graphs are not continuous and not differentiable at \(x=-1\).
For the graph on the left, the function is not defined at \(x=-1\) and therefore is not continuous and not differentiable at \(x=-1\). The function is continuous and differentiable everywhere else.
For the graph on the right, the function has a jump discontinuity at \(x=-1\) (i.e., \(\lim\limits_{x\to-1} f(x)\) does not exist) and therefore is not continuous and not differentiable at \(x=-1\). The function is continuous and differentiable everywhere else.
Long Text Description
There are two graphs displayed. The first graph corresponds to a function that increases for x less than -1, is not defined at x = -1 (indicated by an open circle at the point (-1,1.6)), decreases between x-1 and x=1, and then increases after x = 1. The graph does not have any sharp corners or vertical tangent lines, but has a discontinuity at x = -1.
The second graph corresponds to a function that increases for x less than -1 until it reaches an open circle at the point (-1,1.6). The value of the function then jumps to 2.6, as indicated by a closed circle at the point (-1,2.6). The curve continues to increase from that point, until x = 1, at which point the function decreases.