Does Differentiability Imply Continuity?
3 Choices of Function
Check exactly one of the three check boxes above to select one of three functions. All three functions include the point (1, -3), and all three are discontinuous at x = 1 with different types of discontinuities.
Does a function have to exist at a point in order to be differentiable there?
Recall that for a function to be differentiable at the point x = a the following limit must exist:
. For the derivative (this limit) to even have a hope of existing then the difference quotient (the fraction without the limit) must exist, and this can only happen if f(a) is defined.
Therefore, a function must exist at x = a for f '(a) to exist.
Does differentiablity imply continuity?
Recall that a function is continuous if and only if . This means the function must be defined, the limit must exist, and they must be the same for the function to be continuous.
We know that the function must exist at x = a for both continuity and differentiablity at the point. Let's investigate what happens when the function is defined at x = a, but the function is not continuous there.
Recall that the difference quotient, , is the slope of the secant line from the point of interest (a, f(a)) to a nearby point, B.
In the app:
Notice that the function graphed in blue is a piecewise function which has a discontinuity at x = 1. It is defined there so that (a, f(a)) = (1, -3). Move B around on the graph of f and see what happens to the difference quotient (slope of the red secant line).
Question
Does the derivative exist at x = 1 for this function?
Question
Does a function have to be continuous at a point in order to be differentiable there?