Differentiable Functions

ReminderSummary

  • A function\(f(x)\)has limit\(l\)as\(x→a\)if and only if\(f(x)\)has a left-hand limit at \(x=a\), has a right-hand limit at \(x=a\), and the left- and right-hand limits are equal. Visually, this means that there can be a hole in the graph at \(x=a\), but the function must approach the same single value from either side of \(x=a\).

  • A function\(f(x)\)is continuous at\(x=a\)wheneve\(f(a)\)is defined,\(f\)has a limit as \(x→a\), and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of f at \(x=a\).

  • A function\(f\)is differentiable a\(x=a\)whenever\(f′(a)\)exists, which means that\(f\)has a tangent line at\((a,f(a))\)and thus\(f\)is locally linear at the value \(x=a\). Informally, this means that the function looks like a line when viewed up close at \((a,f(a)) \)and that there is not a corner point or at \((a,f(a))\).

  • Of the three conditions discussed (having a limit at \(x=a\), being continuous at \(x=a\), and being differentiable at \(x=a\), the strongest condition is being differentiable, and the next strongest is being continuous. In particular, if\(f\) is differentiable at \(x=a\), then\(f\)is also continuous at \(x=a\), and if\(f\)is continuous at \(x=a\), then\(f\)has a limit at \(x=a\).

Example

The tangent to the function f(x) = x \sin(\pi x) (blue curve) through x_{0} = 1.