Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.All continuous functions have derivatives.

Step 1
Given,
Statement: All continuous functions have derivatives.
Step 2
The given statement, "All continuous functions have derivatives." is false.
Example: The absolute value function abs x is continuous everywhere but it is not differentiable at x = 0
Since,
${\displaystyle f\left(x\right)=-x,-\mathrm{\infty }\le x\le 0}$
${\displaystyle =x,0\le x\le \mathrm{\infty }}$
Left hand derivative at x = 0 is, -1 and right hand derivative at x = 0 is 1.
Hence the absolute value function is continuous but not differentiable at x = 0.