Question

If a function is continuous at a point, is it true that it is also differentiable at that point?

Answer 1

False

Explanation:

It is no true that if a function is continuous at a point `x=a` then it will also be differentiable at that point.

Take the example of the function `f(x)=absx` which is continuous at every point in its domain, particularly `x=0`.

We can derive that `f'(x)=absx/x`.

Obviously, `f'(x)` does not exist, but `f` is continuous at `x=0`, so the statement is false.

The converse is true, however. Any differentiable function is necessarily continuous.

Proof:

Let `f` be a differentiable function with derivative `f'` defined at every `x in "D"_f`. Then

`f'(x)=lim_(h→0) (f(x+h)-f(x))/h` is well defined

Now, `f` is continuous iff `lim_(x→c) f(x)=f(c)`. Consider `lim_(x→c)f(x)`

`lim_(x→c) f(x)=lim_(h→0)f(c+h)=lim_(h→0) h(f(c+h)-f(c))/h +f(c)=lim_(h→0)hf'(c)+f(c)=f(c)`

So `f` is continuous, as required. `square`

Answer 2

False.

Explanation:

For example, `f(x)=abs(x)` is continuous at `x=0` but it is not differentiable.