Can a piecewise function be continuous and differentiable?

Sal analyzes a piecewise function to see if it’s differentiable or continuous at the edge point. In this case, the function is both continuous and differentiable.

How do you check continuity and differentiability?

For a function to be differentiable at any point x=a in its domain, it must be continuous at that particular point but vice-versa is not always true. Solution: For checking the continuity, we need to check the left hand and right-hand limits and the value of the function at a point x=a. L.H.L = R.H.L = f(a) = 0.

How do you check continuity of a function?

In calculus, a function is continuous at x = a if – and only if – all three of the following conditions are met:

  1. The function is defined at x = a; that is, f(a) equals a real number.
  2. The limit of the function as x approaches a exists.
  3. The limit of the function as x approaches a is equal to the function value at x = a.

How do you know if it is continuous or discontinuous?

For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. Discontinuities may be classified as removable, jump, or infinite.

How do you know if a piecewise function is one to one?

If the graph of a function f is known, it is easy to determine if the function is 1 -to- 1 . Use the Horizontal Line Test. If no horizontal line intersects the graph of the function f in more than one point, then the function is 1 -to- 1 .

What is the condition of continuity to be and differentiability?

dx = . y2 , if y = f (x). Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f (a) = f (b), where a and b are some real numbers. Then there exists at least one point c in (a, b) such that f ′ (c) = 0.