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# 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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## Answer to a math 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.

Birdie
4.5
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.

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