91Ó°ÊÓ

In understanding differentiability, sharp corners are a key concept. When a function's graph has a **sharp corner** at a point, it indicates that the function is not differentiable there. This occurs because the slope of the tangent line changes abruptly, resulting in an undefined derivative at that point.
For example, consider the absolute value function, \[f(x) = |x|\]This function has a sharp corner at \(x = 0\), where it transitions from a slope of -1 to a slope of 1. Because of this sudden direction change, it's impossible to draw a single tangent line at this point that perfectly matches the graph.
In summary:

• A sharp corner results in non-differentiability.
• The slope of the tangent line is undefined at sharp corners.

Thus, while sharp corners do indeed render a function non-differentiable at a point, remember, they're not the only reason.

A vertical tangent presents another situation where differentiability fails. Unlike sharp corners, where the slope is undefined due to abrupt changes, vertical tangents have a slope that tends to infinity.
Imagine examining a curve that shoots nearly vertically up or down as you approach a particular point. At such points, the tangent line, though existent, rises straight up and thus has an infinite slope.Consider the function:\[f(x) = \sqrt[3]{x}\]At \(x = 0\), the graph of the function has a vertical tangent. The derivative, approaching this point, becomes extremely large and undefined (infinite in magnitude), rendering the function non-differentiable at that spot.
Key points to remember include:

• Vertical tangents lead to an undefined infinite slope.
• These occur when the graph of the function sharply turns inward, making differentiability impossible at those points.

Discontinuity provides yet another insight into the concept of non-differentiability. A function can be non-differentiable at points where it is discontinuous. This means if there's a "jump" or "break" in the graph at a certain point, the function cannot possess a derivative there.
For instance, any point on a step function has a discontinuity that results in the absence of any tangent line, as the graph literally breaks or jumps at those points.
Think of discontinuity in these terms:

• Discontinuities prevent a smooth connection in the graph.
• If a function is not continuous at a point, it cannot be differentiable at that point.

Each of these concepts—sharp corners, vertical tangents, and discontinuities—serve as valid instances where a function can lack differentiability, demonstrating that it's not just sharp corners that lead to such conditions.