Let f, g : R → R be given functions. Suppose that f and g are continuous at c ∈ R.Prove that the functionl(x) := inf{f (x), g(x)}, x ∈ R,is continuous at c.

To prove that the function l(x) := inf{f(x), g(x)}, x ∈ R, is continuous at c, we need to show that for every ε > 0, there exists a δ > 0 such that for all x in R with |x - c| < δ, we have |l(x) - l(c)| < ε.

Let's start by noting that since f and g are continuous at c, for every ε > 0, there exist δ1, δ2 > 0 such that for all x in R with |x - c| < δ1, we have |f(x) - f(c)| < ε, and for all x in R with |x - c| < δ2, we have |g(x) - g(c)| < ε.

Now, let's consider the function l(x). We have two cases to consider:

1. If l(c) = f(c), then for all x in R with |x - c| < δ1, we have |l(x) - l(c)| = |inf{f(x), g(x)} - f(c)| ≤ |f(x) - f(c)| < ε.

2. If l(c) = g(c), then for all x in R with |x - c| < δ2, we have |l(x) - l(c)| = |inf{f(x), g(x)} - g(c)| ≤ |g(x) - g(c)| < ε.

In both cases, we have shown that for every ε > 0, there exists a δ > 0 (specifically, δ = min{δ1, δ2}) such that for all x in R with |x - c| < δ, we have |l(x) - l(c)| < ε. Therefore, the function l(x) := inf{f(x), g(x)}, x ∈ R, is continuous at c.