Understanding the Concept of Differentiability and Continuity in Mathematics | Exploring Functions That Are Continuous but Not Differentiable

Can a function be continuous but NOT differentiable?

Yes, it is possible for a function to be continuous but not differentiable

Yes, it is possible for a function to be continuous but not differentiable. In mathematics, continuity and differentiability are separate concepts.

A function is said to be continuous at a point if it has no jumps, breaks, or holes at that point. In other words, its graph can be drawn without lifting the pencil from the paper. Continuity ensures that there are no abrupt changes in the function, and it implies that the limit of the function at that point exists and equals the value of the function at that point.

On the other hand, differentiability refers to the ability to find the derivative of a function at a given point. A function is differentiable at a point if its derivative exists at that point. The derivative describes how the function is changing, and it gives the slope of the tangent line to the graph of the function at that point.

Now, it is possible for a function to be continuous but not differentiable. One common example is the absolute value function, denoted as f(x) = |x|. This function is continuous everywhere, as its graph does not have any jumps or breaks. However, it is not differentiable at x = 0 because the slope of the tangent line on the left and right side of x = 0 is different (i.e., the left derivative and right derivative do not match). The absolute value function has a sharp corner at x = 0, making it non-differentiable at that point.

In general, whenever a function has a sharp corner, a vertical tangent line, a cusp, or any kind of discontinuity in its graph, it is likely to be continuous but not differentiable at that point.

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