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Bijective Fuction

Contrapositiv...

Question

Contrapositive of the statement :
‘If a function $f$ is differentiable at $a$ , then it is also continuous at $a$ , is :

If a function

f

is not continuous at

a

, then it is not differentiable at

a

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If a function

f

is continuous at

a

, then it is differentiable at

a

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If a function

f

is continuous at

a

, then it is not differentiable at

a

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If a function

f

is not continuous at

a

, then it is differentiable at

a

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Solution

The correct option is A If a function $f$ is not continuous at $a$ , then it is not differentiable at $a$ .
Contrapositive of $p \to q$ is
$=\sim q \to\sim p$
i.e., If a function $f$ is not continuous at $a$ , then it is not differentiable at $a$ .

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Similar questions

Q. Write the converse, inverse and contrapositive of the following statements : "If a function is differentiable then it is continuous".

Q. Given the following statements about a function f: R

\to

R, select the right option.

P : If f(x) is continuous at x = x

_{0}

, then it is also differentiable at x = x

_{0}

Q : If f(x) is continuous at x = x

_{0}

, then it may not be differentiable at x = x

_{0}

.

R : If f(x) is differentiable at x = x

_{0}

, then it is continuous at x = x

_{0}

Q. If a function

f (x)

is differentiable at

x = c

then it is continuous a

x = c

Q. Assertion :

f (x) = {\begin{matrix} x^{2} & for x \leq c a x + b & for x > c \end{matrix}

f (x)

is differentiable at

x = c

, then

a = 2 c

and

b = - c^{2}

Reason: A continuous function is differentiable everywhere.

Q. Let

f, g : R \to R

be two functions defined by

f (x) = {\begin{matrix} x s i n (\frac{1}{x}) & x \neq 0 0 & x = 0 \end{matrix}

, and

g (x) = x f (x)

Statement I :

f

is a continuous function at

x = 0

Statement II :

g

is a differentiable function at

x = 0

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Contrapositive of the statement : ‘If a function f is differentiable at a, then it is also continuous at a, is :

The correct option is A If a function f is not continuous at a, then it is not differentiable at a.Contrapositive of p→q is =∼q→∼p i.e., If a function f is not continuous at a, then it is not differentiable at a.

Contrapositive of the statement :
‘If a function $f$ is differentiable at $a$ , then it is also continuous at $a$ , is :

The correct option is A If a function $f$ is not continuous at $a$ , then it is not differentiable at $a$ .
Contrapositive of $p \to q$ is
$=\sim q \to\sim p$
i.e., If a function $f$ is not continuous at $a$ , then it is not differentiable at $a$ .