Find the Points Where the Function Is Not Differentiable

Questions on the differentiability of functions with emphasis connected piecewise functions are presented along with their answers.

Graphical Meaning of not differentiability.
Which Functions are non Differentiable?
Let f live a purpose whose chart is G. From the definition, the value of the derivative of a occasion f at a certain value of x is equal to the slope of the tan to the graph G. We can say that f is not differentiable for any value of x where a tangent cannot 'exist' or the tan exists but is vertical (hierarchal line has undefined slope, hence undefined derivative).
Downstairs are graphs of functions that are not figuring at x = 0 for varied reasons.
Routine f below is not calculation at x = 0 because there is no tan to the graph at x = 0.(try to draw a tangent at x=0!)

Function g

below is not differentiable at x = 0 because there is no tangent to the graph at x = 0.(try to draw a tangent at x=0!)

Function h

below is not distinguishable at x = 0 because there is a jump in the valuate of the use and also the function is not defined therefore not continuous at x = 0.

Function j

below is non differentiable at x = 0 because IT increases indefinitely (no limit) on each sides of x = 0 and also from its formula is undefinable at x = 0 and therefore non continuous at x=0 .

Function k

below is not differentiable because the tangent at x = 0 is unsloped and therefore its slope which the apprais of the derivative at x =0 is undefinable.

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Theorem:

If a function f is differentiable at x = a, then it is continuous at x = a

Contrapositive of the above theorem:

If function f is not continuous at x = a, then it is not differentiable at x = a.
Unwashed mistakes to avoid: If f is continuous at x = a, then f is differentiable at x = a.
NOTE: Although functions f, g and k (whose graphs are shown preceding) are continuous everyplace, they are non differentiable at x = 0.

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Analytical Proofs of non differentiability

Exercise 1: Show analytically that function f defined below is non differentiable at x = 0.

f(x) = \begin{cases} x^2 &ere; x \textgreater 0 \\ - x &A; x \textless 0 \\ 0 & x = 0 \goal{cases}

Solution to Example 1
Unrivalled way to answer the above question, is to cipher the derivative at x = 0. We start aside finding the limit of the difference quotient. Since function f is defined using different formulas, we need to find the derivative at x = 0 using the left-wing and the right limits.

f'(x) = \lim_{h\to\ 0} \dfrac{f(x+h) - f(x)}{h}

On the left of x = 0 (x < 0), the derivative is calculated as follows

f'(0) = \lim_{h\to\ 0^-} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{ -h - 0}{h} = -1

Along the perpendicular of x = 0 (x > 0), the derivative is calculated as follows

f'(0) = \lim_{h\to\ 0^+} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{h^2 - 0}{h} = \lim_{h\to\ 0} h = 0

The limits to the left and to the decently of x = 0 are non equal consequently f '(0) is indefinable and function f in not differentiable at x = 0.

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Thomas More along Never-ending Functions in Tartar
Persistence Theorems and Their use in Calculus
Questions on Continuity with Solutions.
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Find the Points Where the Function Is Not Differentiable

Source: https://www.analyzemath.com/calculus/continuity/non_differentiable.html