Continuity
A function is continuous at a point if its graph can be drawn without lifting the pen from the paper. More formally, a function f(x) is continuous at a point x = a if the following conditions hold:
- f(a) exists: The function is defined at the point a.
- lim_(x→a) f(x) exists: The limit of the function as x approaches a exists.
- lim_(x→a) f(x) = f(a): The limit of the function as x approaches a is equal to the value of the function at a.
Example: The function f(x) = x² is continuous at x = 2 because:
- f(2) = 4 exists.
- lim_(x→2) x² = 4 exists.
- lim_(x→2) x² = f(2).
Differentiability
A function is differentiable at a point if its derivative exists at that point. The derivative of a function represents the instantaneous rate of change of the function at a given point. A function is differentiable at a point if it is smooth at that point, meaning it has no sharp corners or breaks.
Conditions for Differentiability:
- Continuity: A function must be continuous at a point to be differentiable at that point.
- No Vertical Tangents or Cusps: The function must not have a vertical tangent or a cusp at the point in question. This ensures that the derivative exists and is finite.
Example: The function f(x) = |x| is continuous at x = 0 but not differentiable at x = 0. This is because the function has a sharp corner at x = 0.
Practical Insights:
- Continuity is a necessary but not sufficient condition for differentiability. A function can be continuous at a point but not differentiable at that point.
- Differentiable functions are smoother than continuous functions. This is because the derivative of a differentiable function exists and is finite, which means that the function does not have any sharp corners or breaks.
- Many real-world phenomena can be modeled using differentiable functions. This is because differentiable functions allow us to represent the rate of change of these phenomena.
