How to show that a function is continuous on an interval?
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⬆HERE IS YOUR ANSWER⬅
Let us consider a function f(x).
This function will be continuous at any point c∈D, if for any preassigned ε (>0) there exists a δ (>0) such that
|f(x) - f(c)| < ε for ∀x ∈δ-neighbourhood of c∩D.
Example :
*Every constant function is continuous on |R.*
Let, f(x) = k ∀x∈|R and c be any arbitrary point of |R.
∴|f(x) - f(c)| = |k - k| = 0 < ε for any positive ε.
∴|f(x) - f(c)| < ε ∀x∈|x - c| < δ.
∴f(x) is continuous at c.
c being arbitrary f is continuous on |R.
⬆HOPE THIS HELPS YOU⬅
Let us consider a function f(x).
This function will be continuous at any point c∈D, if for any preassigned ε (>0) there exists a δ (>0) such that
|f(x) - f(c)| < ε for ∀x ∈δ-neighbourhood of c∩D.
Example :
*Every constant function is continuous on |R.*
Let, f(x) = k ∀x∈|R and c be any arbitrary point of |R.
∴|f(x) - f(c)| = |k - k| = 0 < ε for any positive ε.
∴|f(x) - f(c)| < ε ∀x∈|x - c| < δ.
∴f(x) is continuous at c.
c being arbitrary f is continuous on |R.
⬆HOPE THIS HELPS YOU⬅
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