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Question

Which of the following is are true?

A
if f(x) be differentiable at the point x=a. Then the function f(x) is continuous at that point.
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B
If functions f(x) and g(x) are continuous at x=a, then f(x)g(x) is also continuous at x=a
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C
If f(x)g(x) is continuous at x=a then functions f(x) and g(x) are also continuous at x=a.
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D
If the function f(x) is continuous at x=a then the function is also differentiable at x=a.
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Solution

The correct options are
A if f(x) be differentiable at the point x=a. Then the function f(x) is continuous at that point.
B If functions f(x) and g(x) are continuous at x=a, then f(x)g(x) is also continuous at x=a
C If f(x)g(x) is continuous at x=a then functions f(x) and g(x) are also continuous at x=a.
A: For a function to be differentiable at some point then function must be continuous at that point and its derivative must be defined (finite)
So if a function is differentiable.
at x=1f(x) is continuous at x=a
B: If f and g are continuous then according to algebra of continuous functions f,g is also continuous.
C For f,g to be continuous
f and g must be continuous
(refer B)
D: If f(x) is continuous at x=a It necessarily does not have to be differentiable at x=a
f(x) should also be defined in addition to f being continuous (refer A)
Only A,B,C are given, Answer is A,B,C.

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