In Maths T, you already learnt how to prove whether a function is continuous. Now you need to know the relationship between continuity and differentiability.

A **differentiable function** has to be continuous, but it doesn’t mean that a **continuous function** is differentiable. Using logical propositions, it means that if f(x) differentiable, then it is continuous, but not conversely. Normally, the non-differentiability occurs in graphs with

**1. a corner ****2. a vertical tangent line**

**3. a discontinuity 4. at end points **

For piece-wise defined functions, it is easy to see whether a function is differentiable at the joints. If the joints have different gradients for the different sub-functions, then it is definitely not differentiable. However, there should be a formal definition for differentiability. For a number **a** in the domain of the function **f**, we say that **f** is differentiable at **a**_{ }, or that the derivatives of **f** exists at **a** if

exists.

You can go on to prove that both formulas are actually the same thing. Of course, differentiability does not restrict to only points. We could also say that a function is differentiable on an interval **(a, b)** or differentiable everywhere, **(-∞, +∞)**. I’ll give you one example:*Prove that f(x) = |x| is not differentiable at x=0.*

So, f(x) = |x| is not differentiable at x = 0. [proven]

** **

These 2 formulas can be used at different situations, so if one doesn’t work, use the other. Differentiability is not a common question in STPM, but you should still be able to make use of this important information. ☺

## No comments:

## Post a Comment