If a function is not ____ at x=c it is also not ____ at x=c
If a function is not continuous at x=c, it is also not differentiable at x=c
If a function is not continuous at x=c, it is also not differentiable at x=c.
Definition: Continuity of a Function at a Point
A function is said to be continuous at a point if the graph of the function has no breaks, jumps, or holes at that particular point. Formally, a function f(x) is continuous at a point x=c if the following three conditions are satisfied:
1. f(c) is defined.
2. The limit of the function as x approaches c, denoted as lim(x→c) f(x), exists.
3. The limit of the function as x approaches c is equal to the value of the function at x=c: lim(x→c) f(x) = f(c).
Definition: Differentiability of a Function at a Point
A function is said to be differentiable at a point if it has a derivative at that point. Formally, a function f(x) is differentiable at a point x=c if the derivative of the function exists at x=c. Geometrically, differentiability means that the graph of the function has a well-defined tangent line at that point without any abrupt changes or corners.
It is important to note that while all differentiable functions are continuous, not all continuous functions are differentiable. This is because a function may be continuous but have sharp changes or corners at certain points, making it not smooth enough to have a tangent line.
Therefore, if a function is not continuous at x=c, it is also not differentiable at x=c since the conditions for continuity are not met.
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