A piecewise function is considered continuous if there are no sudden jumps or breaks in the function at any point. This means that the function can be drawn without lifting the pen from the paper, and the value of the function at any point is equal to the limit of the function as the input approaches that point.
To prove that a piecewise function is continuous, you must first check that the function is defined at the point in question. Then, you must check that the left and right-hand limits of the function at that point are equal. If both of these conditions are met, then the function is continuous at that point.
Continuity refers to the smoothness of a function, while differentiability refers to the existence of a derivative at a given point. A function can be continuous but not differentiable, meaning it has no sudden breaks or jumps, but a derivative cannot be calculated at a particular point.
To prove that a piecewise function is differentiable, you must first check that the function is continuous at the point in question. Then, you must find the derivative of each piece of the function and ensure that they are equal at the point where the pieces meet. If both of these conditions are met, then the function is differentiable at that point.
Yes, it is possible for a piecewise function to be both continuous and differentiable at every point. This occurs when the function is defined by continuous pieces that have continuous derivatives at their points of intersection. An example of this is a polynomial function defined by multiple pieces with different powers of x, as polynomials are both continuous and differentiable at every point.