a problem related to continuous functions

Question

Let $f$ be a function defined in $[0,1]$ such that $f(0) > 0 > f(1)$. Suppose there is a continuous function $g$ on $[0,1]$ such that $f + g$ is increasing. Prove that the equation $f(x)=0$ has at least one solution in $[0,1]$. I tried to consider another continuous function in order to use the theorem of intermediate values but I failed. Thanks a lot.

Answer 1

An increasing function has only jump discontinuities, with left and right limits at every point: $$ \forall a \in (0, 1] : \lim_{x \to a-} (f(x) + g(x)) \le f(a) + g(a) \, ,\\ \forall a \in [0, 1) : f(a) + g(a) \le \lim_{x \to a+} (f(x) + g(x)) \, . $$ But $g$ is continuous, so that $$ \forall a \in (0, 1] : \lim_{x \to a-} f(x) \le f(a) \, , \\ \forall a \in [0, 1) : f(a) \le \lim_{x \to a+} f(x) \, . $$ Now define $$ c = \sup \{ x \mid f(x) > 0 \} \, . $$ Then $$ c > 0 \implies 0 \le f(c) \, ,\\ c < 1 \implies f(c) \le 0 \, . $$ Since $f(0) > 0 > f(1)$ is given, the only possibility is that $0 < c < 1$ and $f(c) = 0$.