Differentiability implies ___________________
Differentiability implies continuity
Differentiability implies continuity.
In mathematics, if a function is differentiable at a particular point, then it is also necessarily continuous at that point. This means that the function does not have any sudden jumps or discontinuities at that point.
Formally, a function f(x) is said to be differentiable at a point a if the following limit exists:
f'(a) = lim (h->0) [ f(a + h) – f(a) ] / h
where h is a small increment in the x-direction. If this limit exists, then the function is said to be differentiable at point a.
Now, if a function is differentiable at all points in its domain, then it is called a differentiable function. Differentiable functions have many useful properties and can be studied in detail using calculus.
To summarize, differentiability implies continuity, but continuity does not necessarily imply differentiability.
More Answers:
Understanding the Exponentiation of Variables | Explained with x^nUnderstanding the Derivative | Calculating Rates of Change and Slopes in Calculus
Derivative Rules | Sum and Difference of Functions Explained