If a function is continuous, then it is differentiable.
not necessarily
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The statement that "if a function is continuous, then it is differentiable" is actually false. Not all continuous functions are differentiable.
To understand why, we first need to clarify the definitions of continuity and differentiability.
A function is said to be continuous at a point if it doesn't have any sudden jumps or breaks at that point. In other words, it means that the function is defined and has a limit at that point. Continuity is a fundamental concept in calculus and is typically defined using limits.
On the other hand, a function is said to be differentiable at a point if it has a well-defined derivative (slope of the tangent line) at that point. The derivative of a function measures how the function changes as its input (usually denoted as x) changes.
Now, while it is true that every differentiable function is continuous, the converse is not true. There exist continuous functions that are not differentiable at certain points.
One classic example is the function f(x) = |x|, also known as the absolute value function. This function is continuous on the entire real number line, but it is not differentiable at x = 0. At that point, the graph has a sharp "corner" and the tangent line does not have a well-defined slope.
Another example is the function f(x) = √x, the square root function. This function is continuous for x ≥ 0 but not differentiable at x = 0. The graph of the function has a sharp corner at x = 0.
In general, if a function has a sharp corner, a vertical tangent line, or a jump discontinuity, it is not differentiable at that point.
To conclude, while all differentiable functions are continuous, not all continuous functions are differentiable. It is crucial to understand the definitions and properties of continuity and differentiability when studying calculus.