Differentiability and Continuity Differentiability measures how a function's rate of change changes with respect to its input. It tells us how quickly th...
Differentiability measures how a function's rate of change changes with respect to its input. It tells us how quickly the function is changing at any point in its domain.
Continuity defines the relationship between a function's graph and its derivative. A function is differentiable if its derivative exists at every point in its domain. This means that the graph of the function can be drawn with a smooth curve without any breaks or jumps.
Key connection between differentiability and continuity:
Differentiability implies continuity: A function is differentiable if its derivative exists at every point in its domain.
Continuity implies differentiability: A function is continuous if its derivative exists at every point in its domain.
A function can be differentiable but not continuous: A function can be differentiable but have a vertical tangent line at some points in its domain.
A function can be continuous but not differentiable: A function can be continuous but have a sharp corner or jump discontinuity at some points in its domain.
Examples:
Differentiable function: A function like f(x) = x^2 is differentiable at every point.
Continuous function: A function like f(x) = x is continuous at every point.
Differentiable but not continuous function: A function like f(x) = |x| is differentiable but is not continuous at x = 0.
Continuous but not differentiable function: A function like f(x) = x^1/2 is continuous but is not differentiable at x = 0