Continuity and Differentiability of a Function

1. 1 Continuity and Differentiability of a Function

2. 2 6.7 Continuity and Differentiability of a FunctionContinuity of a function; Polynomial and rational functions; Differentiability of a function A continuous function: When a function q=g(v) possesses a limit as v tends to the point N in the domain When this limit is also equal to g(N), i.e., the value of the function at v=N Then the function is continuous in N

3. 3 6.7 Continuity and Differentiability of a Function This rational function is not defined at v = 2 and -2 even though the limit exists as v ? 2 or -2. It is discontinuous and therefore does not have continuous derivatives, i.e., it is not continuous differentiable.

4. 4 6.7 Continuity and Differentiability of a Function This continuous function is not differentiable at x = 3 and therefore does not have continuous derivatives, i.e., it is not continuously differentiable

5. 5 6.7 Continuity and differentiability of a function For a function to be continuous differentiable All points in in domain of f defined When the limit concept is applied to the difference quotient at x = x0 as ?x ? 0 from both directions. The continuity condition is necessary but not sufficient. The differentiability condition (smoothness) is both necessary and sufficient for whether f is differentiable, i.e., to move from a difference quotient to a derivative