Question

Let f and g be defined in R and suppose that there exists M > 0 such that |f(x) − f(p)| ≤ M|g(x) − g(p)|, for all x. Prove that if g is continuous in p, then f will also be continuous in p.

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Birdie

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To prove that f is continuous at p, we need to show that for any ε > 0, there exists δ > 0 such that |x - p| < δ implies |f(x) - f(p)| < ε.

Given that |f(x) - f(p)| ≤ M|g(x) - g(p)| for all x, we can rewrite this as |f(x) - f(p)|/|g(x) - g(p)| ≤ M for all x ≠ p.

Since g is continuous at p, we know that for any ε > 0, there exists δ > 0 such that |x - p| < δ implies |g(x) - g(p)| < ε/M.

Now, let's consider the expression |f(x) - f(p)|/|g(x) - g(p)|. If we assume that |x - p| < δ, then we can use the property of g being continuous at p to state that |g(x) - g(p)| < ε/M.

Therefore, we have |f(x) - f(p)|/|g(x) - g(p)| ≤ M, which implies |f(x) - f(p)| < M|g(x) - g(p)| < M(ε/M) = ε.

This shows that for any ε > 0, there exists δ > 0 such that |x - p| < δ implies |f(x) - f(p)| < ε, proving that f is continuous at p.

Therefore, if g is continuous at p, then f is also continuous at p.

Answer: f is continuous at p.

Given that |f(x) - f(p)| ≤ M|g(x) - g(p)| for all x, we can rewrite this as |f(x) - f(p)|/|g(x) - g(p)| ≤ M for all x ≠ p.

Since g is continuous at p, we know that for any ε > 0, there exists δ > 0 such that |x - p| < δ implies |g(x) - g(p)| < ε/M.

Now, let's consider the expression |f(x) - f(p)|/|g(x) - g(p)|. If we assume that |x - p| < δ, then we can use the property of g being continuous at p to state that |g(x) - g(p)| < ε/M.

Therefore, we have |f(x) - f(p)|/|g(x) - g(p)| ≤ M, which implies |f(x) - f(p)| < M|g(x) - g(p)| < M(ε/M) = ε.

This shows that for any ε > 0, there exists δ > 0 such that |x - p| < δ implies |f(x) - f(p)| < ε, proving that f is continuous at p.

Therefore, if g is continuous at p, then f is also continuous at p.

Answer: f is continuous at p.

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