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Continuity refers to a function's ability to be drawn without any breaks or gaps. It means that the values of the function remain close together as the input values change. Differentiability, on the other hand, refers to a function's ability to have a well-defined derivative at a specific point. In other words, it is the smoothness of a function at a particular point.
A function is continuous if it satisfies three conditions: the function is defined at a point, the limit of the function exists at that point, and the limit and the function value at that point are equal. In other words, the left and right-hand limits of the function must equal the function value at that point.
Continuity refers to a function's behavior at a specific point, while uniform continuity refers to a function's behavior over an entire interval. A function is uniformly continuous if the difference between two input values does not affect the difference between their corresponding output values. This means that the function's rate of change is consistent throughout the interval.
A function is differentiable if its derivative exists at a specific point. The derivative of a function is the slope of the tangent line at that point. To determine if a function is differentiable, you can use the limit definition of the derivative or check if the function satisfies the differentiability conditions.
Continuity is a necessary condition for differentiability. If a function is differentiable at a point, it must also be continuous at that point. However, a continuous function may not necessarily be differentiable. This can happen when there is a sharp turn or corner in the graph of the function, which would cause the derivative to be undefined at that point.
